Key exchange method, key exchange system, communication device and storage medium therefore

ABSTRACT

Plurality of users share a common key while permitting dynamic member change and computational complexity required for key exchange is reduced. The first key generation unit 212 of the communication devices Ui computes Ri and ci, or ci based on a twisted pseudo-random function. A session ID generation unit 113 of a key distribution device S generates sid based on a target-collision resistant hash function and transmits sid to the communication devices Ui. A second key generation unit 214 of the communication devices Ui computes Ti based on a pseudo-random function. A third key generation unit 115 of the key distribution device S computes k′ and T′i based on the twisted pseudo-random function. A session key generation unit 217 of the communication devices Ui generates the common key K2 based on a pseudo-random function.

TECHNICAL FIELD

The present invention relates to an application of an information security technology, and especially relates to a key exchange technology by which a plurality of users forming a group share a common key.

The present application claims priority based on Japanese Patent Application 2016-083663 which was filed in Japan on Apr. 19, 2016 and contents thereof are incorporated herein.

BACKGROUND ART

The key exchange technology by which a plurality of users forming a group share a common key has been conventionally proposed. An architecture of an information system for realizing such key exchange technology is described in “Suvo Mittra, “Iolus: a framework for scalable secure multicasting”, SIGCOMM '97, pp. 277-288, 1997” (referred to below as Non-patent literature 1), for example. An algorithm for such key exchange technology is described in “Scalable Multicast Key Distribution”, [online], [retrieved on Mar. 4, 2016], internet <URL:https://tools.ietforg/html/rfc1949>” (referred to below as Non-patent literature 2), for example.

SUMMARY OF THE INVENTION

In the related-art technologies described in Non-patent literatures 1 and 2, since users who share a common key need to be registered in advance, it is impossible for a plurality of users to share the common key while permitting dynamic member change. Further, since the whole computational complexity required for key exchange is O(log n) when the number of users is denoted as n, there is a problem that the computational complexity for the key exchange is increased along with increase of the number of users.

An object of the present invention is to provide a key exchange technology which enables a plurality of users to share a common key while permitting dynamic member change and enables reduction in computational complexity required for key exchange.

In order to solve the above-described problem, a key exchange method according to the present invention is the key exchange method for a case where, in a key exchange system which includes a key distribution device S and n+k pieces (here, n is an integer which is 2 or larger and k is an integer which is 1 or larger) of communication devices U_(i) (i=1, . . . , n+k), communication devices U_(n+1), . . . , U_(n+k) newly join a session established by communication devices U₁, . . . , U_(n), in which ∥ is a concatenation operator, U₁ is one piece of representative communication device which is selected from the communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k), a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S are stored in a storage of the key distribution device S, a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) are stored in a storage of the communication devices U_(i) (i=1, . . . , n+k), and further, information r generated in the session established by the communication devices U₁, . . . , U_(n) is stored in a storage of the communication devices U₁, . . . , U_(n), exchange method including: a first key generation step in which the communication devices U_(i) (i=1, n, . . . , n+k) generate r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, compute R_(i)=g^(ri) and c_(i)=g^(ki)h^(si), and transmit (R_(i), c_(i)) to the key distribution device S, and the communication devices U_(i) (i=2, . . . , n−1) generate k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, compute c_(i)=g^(ki)h^(si), and transmit c_(i) to the key distribution device S; a session ID generation step in which the key distribution device S generates sid by using c₁, . . . , c_(n+k) based on a target-collision resistant hash function and transmits, to the communication devices U_(i), (sid, R_(i−1)) with respect to i=1, 2, sid with respect to i=3, . . . , n−2, (sid, R_(i+1)) with respect to i=n−1, n, and (sid, R_(i−1), R_(i+1)) with respect to i=n+1, . . . , n+k (here, R₀=R_(n+K) and R_(n+k+1)=R₁); a second key generation step in which the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on a pseudo-random function, generates K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function, computes T₁ by an exclusive OR of K₁ ^(l) and K₁ ^(r), computes T′ by an exclusive OR of K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S, the communication device U₂ generates K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function, generates K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function, computes T₂ by an exclusive OR of K₂ ^(l) and K₂ ^(r), and transmits (k₂, s₂, T₂) to the key distribution device S, the communication devices U_(i) (i=3, . . . , n−2) transmit (k_(i), s_(i)) to the key distribution device S, the communication device U_(n−1) generates K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function, generates K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function, computes T_(n−1) by an exclusive OR of K_(n−1) ^(l) and K_(n−1) ^(r), and transmits (k_(n−1), s_(n−1), T_(n−1)) to the key distribution device S, the communication device U_(n) generates K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function, generates K_(n) ^(r) by using (sid, R_(n+1) ^(rn)) based on the pseudo-random function, computes T_(n) by an exclusive OR of K_(n) ^(l) and K_(n) ^(r), and transmits (k_(n), s_(n), T_(n)) to the key distribution device S, and the communication devices U_(i) (i=n+1, . . . , n+k) generate K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function, generate K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function, compute T_(i) by an exclusive OR of K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S; a third key generation step in which the key distribution device S generates k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes k′ by an exclusive OR of k₂, . . . , k_(n+k), k_(s), computes T′, by an exclusive OR of T₁, . . . , T_(i−1) with respect to i=2, . . . , n+k (here, T_(i) is nil with respect to i=3, . . . , n−1), transmits k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (i=2, . . . , n+k); a first session key generation step in which the communication devices U_(i) (i=2, . . . n, n+k) compute K₁ ^(l) by an exclusive OR of T′_(i) and K_(i) ^(l) and compute k₁∥s₁ by an exclusive OR of T′ and K₁ ^(l), and the communication devices U_(i) (i=3, . . . , n−1) compute K₁ ^(l) by an exclusive OR of T′_(i) and g^(r) and compute k₁∥s₁ by the exclusive OR of T′ and K₁ ^(l); and a second session key generation step in which the communication devices U_(i) (i=1, . . . , n+k) generate a common key K₂ by using sid and an exclusive OR of k′ and k₁ based on the pseudo-random function.

Further, a key exchange method according to the present invention is the key exchange method for a case where, in a key exchange system which includes a key distribution device S and n pieces (here, n is an integer which is 2 or larger) of communication devices U_(i) (i=1, . . . , n), communication devices U_(j1), . . . , U_(jm) leave from a session established by communication devices U₁, . . . , U_(n), in which R={U_(j1), . . . , U_(jm)} is a subset of {U₁, . . . , U_(n)} and N={U_(j1−1), U_(j1+1), . . . , U_(jm−1), U_(jm−1)} is a subset of {U₁, . . . , U_(n)}, ∥ is a concatenation operator, U₁ (∈N) is one piece of representative communication device which is selected from N, a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S are stored in a storage of the key distribution device S, and a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and information H_(i) ^(l) and H_(i) ^(r) generated in the session established by the communication devices U₁, . . . , U_(n) are stored in a storage of the communication devices U_(i) (i=1, . . . , n), the key exchange method including: a first key generation step in which the communication devices U_(i)(∈N) generate r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, compute R_(i)=g^(ri) and c_(i)=g^(ki)h^(si), and transmit (R_(i), c_(i)) to the key distribution device S, and the communication devices U_(i)(∈({U₁, . . . , U_(n)}−R)−N) generate k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, compute c_(i)=g^(ki)h^(si), and transmit c_(i) to the key distribution device S; a session ID generation step in which the key distribution device S generates sid by using {c_(i)|i satisfies U_(i)∈{U₁, . . . , U_(n)}−R} based on a target-collision resistant hash function and transmits, to the communication devices U_(i), (sid, R_(j)) with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (here, j is a minimum index which satisfies U_(j)∈N and j>i), (sid, R_(j′)) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i), and sid with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N; a second key generations step in which the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on a pseudo-random function, generates K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes T₁ by an exclusive OR of K₁ ^(l) and K₁ ^(r), computes T′ by an exclusive OR of K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(j′)=U_(n−1) are satisfied, generates K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates K₁ ^(r) by using (sid, H₁ ^(r)) based on the pseudo-random function, computes T₁ by the exclusive OR of K₁ ^(l) and K₁ ^(r), computes T′ by the exclusive OR of K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j′)=U_(n−1) and U₂∈N are satisfied, and generates K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function, generates K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes T₁ by the exclusive OR of K₁ ^(l) and K₁ ^(r), computes T′ by the exclusive OR of K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(n)∈N are satisfied, the communication devices U_(i) (i satisfies U_(i)∈N and U_(i+1)∈R (here, i is not 1)) generate K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generate K_(i) ^(r) by using (sid, R_(j) ^(ri)) based on the pseudo-random function, compute T_(i) by an exclusive OR of K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S, the communication devices U_(i) (i satisfies U_(i)∈N and U_(i−1)∈R (here, i is not 1)) generate K_(i) ^(l) by using (sid, R_(j′) ^(ri)) based on the pseudo-random function, generate K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, compute T_(i) by the exclusive OR of K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S, and the communication devices U (i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (here, i is not 1)) generate K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generate K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, compute T_(i) by the exclusive OR of K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S; a third key generation step in which the key distribution device S generates k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes k′ by an exclusive OR of {k_(i)|i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}} and k_(s), computes T′_(i) by an exclusive OR of T₁, . . . , T_(j), . . . , T_(i−1) with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R (here, T_(j) is nil with respect to j which satisfies U_(j)∈R), transmits k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i)(∈({U₁, . . . , U_(n)}−R)−{U₁}); a first session key generation step in which the communication devices U_(i)(∈({U₁, . . . , U_(n)}−R)−{U₁}) compute K₁ ^(l) by an exclusive OR of T′₁ and K_(i) ^(l) and compute k₁∥s₁ by an exclusive OR of T′ and K₁ ^(l); and a second session key generation step in which the communication devices U_(i)(∈{U₁, . . . , U}−R) generate the common key K₂ by using sid and the exclusive OR of k′ and k₁ based on the pseudo-random function.

According to the present invention, a plurality of users can share a common key while permitting dynamic member change. Computational complexity required for key exchange is the constant number of times which is the number of users, that is, O(1), thus being reduced more than the related art.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates the functional configuration of a key exchange system.

FIG. 2A illustrates the functional configuration of a key distribution device.

FIG. 2B illustrates the functional configuration of a communication device.

FIG. 3 illustrates a processing flow (system set-up) in a key exchange method.

FIG. 4 illustrates a processing flow (session key distribution) in the key exchange method.

FIG. 5 illustrates a processing flow (user addition) in the key exchange method.

FIG. 6 illustrates a processing flow (user deletion) in the key exchange method.

DETAILED DESCRIPTION OF THE EMBODIMENT

Prior to the description of an embodiment, the notation in this specification will be described.

To select an element m at random from Set which is a certain set is expressed as m∈_(R)Set.

To output y based on ALG, which is a certain algorithm, with respect to an input x and a random number r is expressed as y←ALG(x;r). Here, in the case where ALG is a deterministic algorithm, the random number r is nil.

|⋅| denotes the bit length of a value ⋅.

κ denotes a security parameter.

F={F_(κ): Dom_(κ)×FS_(κ)→Rng_(κ)}_(κ) is a family of functions including a definition range {Dom_(κ)}_(κ), a key space {FS_(κ)}_(κ), and a value range {Rng_(κ)}_(κ). In this case, if a function F_(κ) and a true random function RF_(κ): Dom_(κ)→Rng_(κ) cannot be distinguished with respect to a person D to be identified at arbitrary polynomial time, F={F_(κ)}_(κ) is called a family of pseudo-random functions. A specific example of the pseudo-random function is described in Reference literature 1 “O. Goldreich, “Modern Cryptography, Probabilistic Proofs and Pseudo-randomness”, Springer-Verlag Tokyo, 2001”, for example.

H={H_(κ): Dom_(κ)→Rlig_(κ)}_(κ) is a family of hash functions including the definition range {Dom_(κ)}_(κ) and the value range {Rng_(κ)}_(κ). In this case, if x′ (≠x) which satisfies H_(κ)(x)=H_(κ)(x′) when x∈_(R)Dom_(κ) is provided cannot be found with respect to an attacker A at arbitrary polynomial time, H={H_(κ)}_(κ) is called a family of target-collision resistant hash functions. A specific example of the target-collision resistant hash function is described in Reference literature 2 “J. A. Buchmann, “Introduction to Cryptography—Edition 3”, Maruzen Publishing Co., Ltd., 2007”, for example.

Public key encryption algorithms are defined as (Gen, Enc, Dec). In a key generation algorithm Gen, a security parameter κ is an input and a public key pk and a secret key sk are outputs. In an encryption algorithm Enc, the public key pk and a plaintext in are inputs and a cipher text CT is an output. In the decryption algorithm Dec, the secret key sk and the cipher text CT are inputs and the plaintext m is an output. A specific example of the public key encryption algorithm is described in Reference literature 2 mentioned above, for example.

Message authentication code algorithms are defined as (MGen, Tag, Ver). In a MAC key generation algorithm MGen, the security parameter κ is an input and a MAC key ink is an output. In the tag generation algorithm Tag, the MAC key ink and the plaintext in are inputs and the authentication tag σ is an output. In a verification algorithm Ver, the MAC key mk, the plaintext in, and the authentication tag σ are inputs, and 1 is outputted when the authentication tag σ is correct, while 0 is outputted when the authentication tag σ is incorrect. A specific example of the message authentication code algorithm is described in Reference literature 2 mentioned above, for example.

Functional encryption algorithms are defied as (Setup, Der, FEnc, FDec). In a setup algorithm Setup, the security parameter κ is an input and a master secret key msk and a public parameter Params are outputs. In a key derivation algorithm Der, the public parameter Params, the master secret key msk, and attribution A are inputs and a user secret key usk is an output. In an encryption algorithm FEnc, the public parameter Params, an access structure P, and the plaintext in are inputs and the cipher text CT is an output. In a decryption algorithm FDec, the user secret key usk and the cipher text CT are inputs and the plaintext m is outputted if the attribution A satisfies the access structure P. A specific example of the functional encryption algorithm is described in Reference literature 3 “D. Boneh, A. Sahai, and B. Waters, “Functional encryption: definitions and challenges”, TCC, Lecture Notes in Computer Science, vol. 6597, pp. 253-273, 2011”, for example.

A function tPRF: {0, 1}^(κ)×FS_(κ)×{0, 1}^(κ)×FS_(κ)→Rng_(κ) is called a twisted pseudo-random function, and tPRF(a,a′,b,b′):=F _(κ)(a,b)⊕F _(κ)(b′,a′)

is defined by using the pseudo-random function F_(κ). Here, a, b′∈{0, 1}^(κ) and a′, b∈FS_(κ). A specific example of the twisted pseudo-random function is described in Reference literature 4 “Kazuki Yoneyama, “One-Round Authenticated Key Exchange with Strong Forward Secrecy in the Standard Model against Constrained Adversary”, IEICE Transactions, vol. E96-A, no. 6, pp. 1124-1138, 2013”, for example.

An embodiment of the present invention will be detailed below. Components having identical functions in the drawings will be denoted by identical reference characters and duplicate description thereof will be omitted.

<System Structure>

As illustrated in FIG. 1, the key exchange system according to the embodiment includes a key distribution device 1 and N (≥2) pieces of communication devices 2 ₁, . . . , 2 _(N). In this embodiment, the key distribution device 1 and the communication devices 2 ₁, . . . , 2 _(N) are respectively connected to a communication network 3. The communication network 3 is a communication network adopting the circuit switching system or the packet switching system which is configured so that the key distribution device 1 can communicate with each of the communication devices 2 ₁, . . . , 2 _(N). In this embodiment, the communication devices 2 ₁, . . . , 2 _(N) do not have to be able to communicate with each other. The communication network 3 does not have to be a communication path in which safety is ensured but an internet or the like, for example, can be employed as the communication network 3.

The key distribution device 1 includes a storage 100, a setup unit 101, a public key generation unit 102, a secret string generation unit 103, a user key transmission unit 111, a session ID generation unit 113, an authentication tag verification unit 114, a third key generation unit 115, and an authentication tag generation unit 116, as illustrated in FIG. 2A. The communication device 2 includes a storage 200, a public key generation unit 202, a secret string generation unit 203, a user key reception unit 211, a first key generation unit 212, a second key generation unit 214, an authentication tag generation unit 215, an authentication tag verification unit 216, and a session key generation unit 217, as illustrated in FIG. 2B. The key distribution device 1 and the communication devices 2 ₁, . . . , 2 _(N) perform processing of each step illustrated in FIGS. 3, 4, 5, and 6, realizing the key exchange method according to the embodiment.

The key distribution device 1 and the communication devices 2 ₁, . . . , 2 _(N) are special devices which are configured such that a special program is read into well-known or dedicated computers including a central processing unit (CPU), a main storage device (random access memory: RAM), and the like. Each device executes processing under the control of the CPU, for example. Data inputted into each device and data obtained through each processing are stored in the main storage device, for example, and the data stored in the main storage device is read onto the CPU as appropriate to be used for other processing. At least part of processing units included in each device may be composed of hardware such as an integrated circuit.

The storage 100 included in the key distribution device 1 and the storage 200 included in the communication devices 2 ₁, . . . , 2 _(N) may be composed of a main storage device such as a random access memory (RAM), an auxiliary storage device composed of a hard disk, an optical disk, or a semiconductor memory element such as a flash memory, or middleware such as a relational database and a key value store. Since each storage stores secret information, it is preferable that each storage is a storage device having tamper resistance (a SIM card, for example).

<System Setup>

A processing procedure for system setup in the key exchange method according to the embodiment will be described with reference to FIG. 3.

In the following description, symbols will be defined as the following. S denotes the key distribution device 1 and U_(i)(i∈{1, . . . , N}) denote N pieces of communication devices 2 ₁, . . . , 2 _(N). G denotes a multiplication cyclic group of a prime number order p of κ bits. Each of g and h denotes a generation source of the group G H: {0, 1}*→{0, 1}K denotes a target-collision resistant hash function. tPRF: {0, 1}^(κ)×FS_(κ)×{0, 1}^(κ)×FS_(κ)→Z_(p) and tPRF′: {0, 1}^(κ)×FS_(κ)×{0, 1}^(κ)×FS_(κ)→FS_(κ) denote twisted pseudo-random functions. F: {0, 1}^(κ)×G→Z_(p) ², F′: {0, 1}^(κ)×Z_(p)→FS_(κ), and F″: {0, 1}^(κ)×FS_(κ)→{0, 1}^(κ) denote pseudo-random functions.

In step S101, the setup unit 101 of the key distribution device S generates the public parameter Params and the master secret key msk based on the setup algorithm Setup for functional encryption. The setup unit 101 transmits the public parameter Params to each of the communication devices U₁, . . . , U_(N). The master secret key msk is stored in the storage 100.

In step S102, the public key generation unit 102 of the key distribution device S generates a combination of the public key pk_(S) and the secret key sk_(S) of the key distribution device S based on the key generation algorithm Gen for public key encryption. The public key pk_(S) of the key distribution device S is made public by using a public key infrastructure or the like, for example. The secret key sk_(S) of the key distribution device S is stored in the storage 100.

In step S202, the public key generation unit 202 of the communication devices U_(i) generates a combination of the public key pk_(i) and the secret key sk_(i) of the communication device U_(i) based on the key generation algorithm Gen for public key encryption. The public key pk_(i) of the communication devices U_(i) is made public by using a public key infrastructure or the like, for example. The secret key sk_(i) of the communication devices U_(i) is stored in the storage 200.

In step S103, the secret string generation unit 103 of the key distribution device S generates secret strings (st_(S), st′_(S)) used in the twisted pseudo-random function as st_(S)∈_(R)FS_(κ) and st′_(S)∈{0, 1}_(κ). The secret strings (st_(S), st′_(S)) are stored in the storage 100.

In step S203, the secret string generation unit 203 of the communication devices U_(i) generates secret strings (st_(i), st′_(i)) used in the twisted pseudo-random function as st_(i)∈_(R)FS_(κ) and st′_(i)∈{0, 1}_(κ). The secret strings (st_(i), st′_(i)) are stored in the storage 200.

In step S104, the key distribution device S acquires public keys pk₁, . . . , pk_(N) of respective communication devices U₁, . . . , U_(N) from a public key infrastructure or the like, for example, so as to store the public keys pk₁, . . . , pk_(N) in the storage 100.

In step S204, the communication devices U_(i) acquire the public key pk_(S) of the key distribution device S from a public key infrastructure or the like, for example, so as to store the public key pk_(S) in the storage 200. Further, the communication devices U₁ store the public parameter Params, which is received from the key distribution device S, in the storage 200.

<Session Key Distribution>

A processing procedure for session key distribution in the key exchange method according to the embodiment will be described with reference to FIG. 4.

In the following description, it is assumed that arbitrary n (≤N) pieces of communication devices U_(i)(i∈{1, . . . , n}) among N pieces of communication devices 2 ₁, . . . , 2 _(N) share a session key SK. In the case where S and U_(i) are inputs of each algorithm, S and U_(i) are identifiers for uniquely specifying respective devices.

In step S111, in the case where a session is started by the communication devices U_(i) (i∈{1, . . . , n}) and the session is the first session in a time frame TF of the communication devices U the user key transmission unit 111 of the key distribution device S generates a user secret key usk_(i)←Der(Params, msk, A_(i)) of the communication devices U_(i) based on the key derivation algorithm Der for functional encryption with current time and attribution respectively used as time and A_(i)=(U_(i), time). Further, the user key transmission unit 111 generates a MAC key mk_(i)←MGen of the communication devices U_(i) based on the key generation algorithm MGen for a message authentication code. Then, the user key transmission unit 111 encrypts the user secret key usk_(i) and the MAC key mk_(i) by using the public key pk_(i) of the communication devices U_(i) based on the encryption algorithm Enc for public key encryption so as to generate the cipher text CT_(i)←Enc_(pki)(usk_(i), mk_(i)). The user key transmission unit 111 transmits the cipher text CT_(i) to each of the communication devices U_(i).

In step S211, the user key reception unit 211 of the communication devices U_(i) decrypts the cipher text CT_(i), which is received from the key distribution device S, by using the secret key sk_(i) of the communication devices U_(i) based on the decryption algorithm Dec for public key encryption so as to obtain a user secret key and a MAC key (usk_(i), mk_(i))←Dec_(ski)(CT_(i)). The user key reception unit 211 stores the user secret key usk_(i) and the MAC key mk_(i) in the storage 200.

In step S212, the first key generation unit 212 of the communication devices U_(i) generates ˜r_(i)∈_(R){0, 1}^(κ), ˜r′_(i)∈_(R)FS_(κ), ˜k_(i)∈_(R) {0, 1}^(κ), ˜k′_(i)∈_(R)FS_(κ), ˜s_(i)∈_(R){0, 1}^(κ), and ˜s′_(i)∈_(R)FS_(κ) and computes r_(i)=tPRF(˜r_(i), ˜r′_(i), st_(i), st′_(i)), k_(i)=tPRF(˜k_(i), ˜k′_(i), st_(i), st′_(i)), and s_(i)=tPRF(˜s_(i), ˜s′_(i), st_(i), st′_(i)) based on the twisted pseudo-random function tPRF. Further, the first key generation unit 212 computes R_(i)=g^(ri) and c_(i)=g^(ki)h^(si). Then, the first key generation unit 212 transmits (R_(i), c_(i)) to the key distribution device S.

In step S112, the key distribution device S receives (R_(i), c_(i)) from the communication devices U_(i). At this time, the key distribution device S stands by until the key distribution device S receives (R₁, c₁), (R_(n), c_(n)) respectively from all of the communication devices U₁, . . . , U_(n).

In step S113, the session ID generation unit 113 of the key distribution device S generates sid=H(c₁, . . . , c_(n)) by using c₁, . . . , c_(n), which are respectively received from the communication devices U₁, . . . , U_(n), based on the target-collision resistant hash function H. Further, one piece of communication device is selected as a representative from n pieces of communication devices U₁, . . . , U_(n). A representative may be arbitrarily selected. For example, a predetermined communication device with the highest priority or a communication device which has started a session most recently may be selected. It is assumed that the communication device U₁ is selected, and U₁ is called a representative communication device in this example. Further, n−1 pieces of communication devices U_(j)(j∈{2, . . . , n}) other than the representative communication device U₁ are called general communication devices. The session ID generation unit 113 computes α and β as the following formulas and transmits (sid, R_(α), R_(β)) to each of the communication devices U_(i).

$\alpha = \left\{ {\begin{matrix} {i - 1} & {{{{if}\mspace{14mu} i} - 1} \geq 1} \\ {i - 1 + n} & {{{{if}\mspace{14mu} i} - 1} < 1} \end{matrix},{\beta = \left\{ \begin{matrix} {i + 1} & {{{{if}\mspace{14mu} i} + 1} \leq n} \\ {i + 1 - n} & {{{{if}\mspace{14mu} i} + 1} > n} \end{matrix} \right.}} \right.$

In step S213, each of the communication devices U_(i) receives (sid, R_(α), R_(β)) from the key distribution device S. The communication devices U_(i) execute the following processing as soon as the communication devices U_(i) receive (sid, R_(α), R_(β)). In the case where i=2, . . . , n holds, that is, the case where the communication devices U_(i) are the communication devices U_(j) (i=j), the processing is progressed to step S214 _(j). In the case where i=1 holds, that is, the case where the communication device U_(i) is the representative communication device U₁, the processing is progressed to step S214 ₁.

In step S214 _(j), the second key generation unit 214 of the general communication devices U_(j) (j∈{2, . . . , n}) generates K_(i) ^(l) by using (sid, R_(α) ^(rj)) based on the pseudo-random function F and generates K_(j) ^(r) by using (sid, R_(β) ^(rj)) based on the pseudo-random function F so as to compute T_(j) by an exclusive OR of K_(j) ^(l) and K_(j) ^(r), as the following formulas. K ^(l) _(j) =F(sid,R ^(r) ^(j) _(α)), K ^(r) _(j) =F(sid,R ^(r) ^(j) _(β)), T _(j) =K ^(l) _(j) ⊕K ^(r) _(j).

In step S215 _(j), the authentication tag generation unit 215 of the general communication devices U_(j) generates an authentication tag σ_(j)=Tag_(mkj)(R_(j), c_(j), R_(α), R_(β), k_(j), s_(j), T_(j), U_(j), sid) by using the MAC key mk_(j) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(j), s_(j), T_(j), σ_(j)) to the key distribution device S.

In step S214 ₁, the second key generation unit 214 of the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n) ^(r1)) based on the pseudo-random function F so as to compute T₁ by an exclusive OR of K₁ ^(l) and k_(i)∥s₁, as the following formulas. Here, ∥ denotes a concatenation operator. K ₁ ^(l) =F(sid,R _(n) ^(r) ¹ ), T ₁ =K ₁ ^(l) ⊕k ₁ ∥s ₁

In step S215 ₁, the authentication tag generation unit 215 of the representative communication device U₁ generates an authentication tag σ_(i)=Tag_(mk1)(R₁, c₁, R_(n), R₂, T₁, U₁, sid) by using the MAC key mk₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (T₁, σ₁) to the key distribution device S.

In step S114 _(j), the authentication tag verification unit 114 of the key distribution device S receives (k_(j), s_(j), T_(j), σ_(j)) from the general communication devices U_(j) with respect to j=2, . . . , n and verifies Ver_(mkj)(R_(j), c_(j), R_(α), R_(β), k_(j), s_(j), T_(j), U_(j), sid, σ_(j)) by using the MAC key mk_(j) of the general communication devices U_(j) based on the verification algorithm Ver for a message authentication code. When the authentication tag σ_(j) is invalid, the authentication tag verification unit 114 ends the session of the general communication devices U_(j). Further, the authentication tag verification unit 114 verifies whether or not c_(j)=g^(kj)h^(sj) is satisfied with respect to j=2, n. When c_(j)=g^(kj)h^(sj) is not satisfied, the authentication tag verification unit 114 ends the session of the general communication devices U_(j).

In step S114 ₁, the authentication tag verification unit 114 of the key distribution device S receives (T₁, σ₁) from the representative communication device U₁ and verifies Ver_(mk1)(R₁, c₁, R_(n), R₂, T₁, U₁, sid, σ₁) by using the MAC key mk₁ of the representative communication device U₁ based on the verification algorithm Ver for a message authentication code. When the authentication tag σ₁ is invalid, the authentication tag verification unit 114 ends the session of the representative communication device U₁.

In step S115 a, the third key generation unit 115 of the key distribution device S generates ˜k_(S)∈_(R){0, 1}^(κ), ˜k′_(S)∈_(R)FS_(κ), ˜K₁∈_(R){0, 1}^(κ), and ˜K′₁∈_(R)FS_(κ) so as to compute k_(S)=tPRF(˜k_(S), ˜k′_(S), st_(S), st′_(S)) and K₁=tPRF(˜K₁, ˜K′₁, st_(S), st′_(S)) based on the twisted pseudo-random function tPRF. Further, the third key generation unit 115 computes k′ by the following formula. k′=(⊕_(2≤j≤n) k _(j))⊕k _(S)

In step S115 b, the third key generation unit 115 of the key distribution device S computes T′_(j) with respect to j=2, . . . , n by the following formula. T′ _(j)=⊕_(1≤i≤j−1) T _(i)

In step S115 c, the third key generation unit 115 of the key distribution device S encrypts a common key K₁ with respect to i=1, . . . , n based on the encryption algorithm FEnc for functional encryption with the access structure P_(i)=(ID=U_(i)){circumflex over ( )}(time∈TF) so as to generate a cipher text CT′_(i)=FEnc(Params, P_(i), K₁). Here, ID is a predicate variable representing a communication device and TF is a predicate variable representing a time frame of the communication device.

In step S116 _(j), the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(j)=Tag_(mkj)(R_(j), c_(j), R_(α), R_(β), k_(j), s_(j), T_(j), U_(j), sid, c₁, k′, T′_(j), T₁, CT′_(j)) with respect to j=2, n by using the MAC key mk_(j) of the general communication devices U_(j) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(j), T₁, CT′_(j), σ′_(j)) to the general communication devices U_(j).

In step S116 ₁, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′₁=Tag_(mk1)(R₁, c₁, R_(n), R₂, T₁, U₁, sid, k′, CT′₁) by using the MAC key mk₁ of the representative communication device U₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (k′, CT′₁, σ′_(i)) to the representative communication device U₁.

In step S216 _(j), the authentication tag verification unit 216 of the general communication devices U_(j)(j∈{2, . . . , n}) receives (c₁, k′, T′_(j), T₁, CT′_(j), σ′_(j)) from the key distribution device S and verifies Ver_(mkj)(R_(j), c_(j), R_(α), R_(β), k_(j), s_(j), T_(j), U_(j) sid, c₁, k′, T′_(j), T₁, CT′_(j), σ′₃) by using the MAC key mk_(j) of the general communication devices U_(j) based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′_(j) is invalid, the authentication tag verification unit 216 ends the session of the general communication devices U_(j). Further, the authentication tag verification unit 216 computes K₁ ^(l) by an exclusive OR of T′_(j) and K_(j) ^(l) and computes k₁∥s₁ by an exclusive OR of T₁ and K₁ ^(l), as the following formulas. K ₁ ^(l) =T′ _(j) ⊕K ^(l) _(j), k ₁ ∥s ₁ =T ₁ ⊕K ₁ ^(l)

Then, the authentication tag verification unit 216 verifies whether or not c₁=g^(k1)h^(s1) is satisfied. When c₁=g^(k1)h^(s1) is not satisfied, the authentication tag verification unit 216 ends the session of the general communication devices U_(j).

In step S216 ₁, the authentication tag verification unit 216 of the representative communication device U₁ receives (k′, CT′₁, σ′₁) from the key distribution device S and verifies Ver_(mk1)(R₁, c₁, R_(n), R₂, T₁, U₁, sid, k′, CT′₁, σ′₁) by using the MAC key mk₁ of the representative communication device U₁ based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′₁ is invalid, the authentication tag verification unit 216 ends the session of the representative communication device U₁.

In step S217, the session key generation unit 217 of the communication devices U_(i) decrypts the common key K₁←FDec_(uski)(CT′_(i), P_(i)) by using the user secret key usk_(i) of the communication devices U_(i) based on the decryption algorithm FDec for functional encryption. Further, the session key generation unit 217 computes a common key K₂ based on the pseudo-random function F′ as the following formula. K ₂ =F′(s id,k′⊕k ₁)

Then, the session key generation unit 217 computes a session key SK based on the pseudo-random function F″ as the following formula. SK=F″(sid,K ₁)⊕F″(sid,K ₂)

According to the key exchange technology of the present invention, the above-described configuration makes it unnecessary to preliminarily register information of users who perform key exchange as the related art, thus enabling a plurality of users to share a common key while permitting dynamic member change. Further, the whole computational complexity required for the key exchange has been O(log n) when the number of users is set as n in the related art, but according the present invention, the whole computational complexity is the constant number of times which is the number of users, that is, O(1), enabling key exchange with smaller computational complexity than the related art.

Here, the processing for guaranteeing integrity by using the message authentication code algorithm, that is, the processing for generation and verification of an authentication tag in S215 _(j), S215 ₁, S114 _(j), S114 ₁, S116 _(j), S116 ₁, S216 _(j), and S216 ₁ may be omitted.

Further, the communication devices U_(i) are configured to generate two keys, that is, the common key K₂ and the session key SK from sid based on the pseudo-random function so as to share these two keys among the communication devices U_(i). However, the communication devices U_(i) may be configured to generate and share only the common key K₂.

A processing procedure of the key exchange method in user addition and user deletion will be described with reference to FIGS. 5 and 6. It is assumed that the session key SK has been already shared among the communication devices U_(i)(i∈1, . . . , n). Each of the communication devices U_(i) stores secret information to be used for user addition and user deletion in the storage 200 after the end of the key exchange processing (after S217).

The representative communication device U_(i) stores secret information H₁ ^(l), H₁ ^(r), and r computed by the following formulas in the storage 200. H ₁ ^(l) =R _(n) ^(r) ^(i) H ₁ ^(r) =R ₂ ^(r) ^(i) r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

Here, F′″: {0, 1}^(κ)×FS_(κ)→Z_(p) is a pseudo-random function.

The general communication devices U_(i)(i∈{2, . . . , n}) store secret information H_(i) ^(l), H_(i) ^(r), and r computed by the following formulas in the storage 200. H _(i) ^(l) =R _(i−1) ^(r) ^(i) H _(i) ^(r) =R _(i+1) ^(r) ^(i) r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

<User Addition>

A processing procedure of user addition in the key exchange method according to the embodiment will be described with reference to FIG. 5.

It is assumed that communication devices U_(n+1), . . . , U_(n+k) (k is an integer which is 1 or larger and n+k is an integer which is N or smaller) newly join a session established by the communication devices U₁, . . . , U_(n). Hereinafter, it is assumed that i∈{1, . . . , n+k} holds.

In step S311, in the case where a session is started by the communication device U_(i) (i=1, . . . , n+k) and the session is the first session in a time frame TF′ of the communication devices U_(i), the user key transmission unit 111 of the key distribution device S generates a user secret key usk_(i)←Der(Params, msk, A_(i)) of the communication device U_(i) based on the key derivation algorithm Der for functional encryption, with current time and attribution respectively used as time and A_(i)=(U_(i), time). Further, the user key transmission unit 111 generates a MAC key mk_(i)←MGen of the communication device U_(i) based on the key generation algorithm MGen for a message authentication code. Then, the user key transmission unit 111 encrypts the user secret key usk_(i) and the MAC key mk_(i) by using the public key pk_(i) of the communication devices U_(i) based on the encryption algorithm Enc for public key encryption so as to generate the cipher text CT_(i)←Enc_(pki)(usk_(i), mk_(i)). The user key transmission unit 111 transmits the cipher text CT_(i) to each of the communication devices U_(i).

Further, in the case where a time frame has been changed from the session established by the communication devices U₁, . . . , U_(n), that is, in the case where TF′ and TF are not equal to each other, the key distribution device S generates ˜K₁∈_(R){0, 1}^(κ) and ˜K′_(i)∈_(R)FS_(κ) and computes K₁=tPRF′(˜K₁, ˜K′₁, st_(S), st′_(S)) based on the twisted pseudo-random function tPRF′. Here, the computation of K₁ is performed by the third key generation unit 115 of the key distribution device S in step S315 a which will be described later, so that the computation does not have to be performed here.

In step S411, the user key reception unit 211 of the communication devices U_(i) (i=1, . . . , n+k) decrypts the cipher text CT_(i), which is received from the key distribution device S, by using the secret key sk_(i) of the communication devices U_(i) based on the decryption algorithm Dec for public key encryption so as to obtain a user secret key and a MAC key (usk_(i), mk_(i))←Dec_(ski)(CT_(i)). The user key reception unit 211 stores the user secret key usk_(i) and the MAC key mk_(i) in the storage 200.

In step S412, in the case of i∈{1}∪[n, n+k], the first key generation unit 212 of the communication device U_(i) generates ˜r_(i)∈_(R){0, 1}^(κ), ˜r′_(i)∈_(R)FS_(κ), ˜k_(i)∈_(R){0, 1}^(κ), ˜k′_(i)∈_(R)FS_(κ), ˜s_(i)∈_(R){0, 1}^(κ), and ˜s′_(i)∈_(R)FS_(κ) so as to compute r_(i)=tPRF(˜r_(i), ˜r′_(i), st_(i), st′_(i)), k_(i)=tPRF(˜k_(i), ˜k′_(i), st_(i), st′_(i)), and s_(i)=tPRF(˜s_(i), ˜s′_(i), st_(i), st′_(i)) based on the twisted pseudo-random function tPRF. Further, the first key generation unit 212 computes R_(i)=g^(ri) and c_(i)=g^(ki)h^(si). Then, the first key generation unit 212 transmits (R_(i), c_(i)) to the key distribution device S.

In the case of i∈[2, n−1], the first key generation unit 212 of the communication devices U_(i) generates ˜k_(i)∈_(R){0, 1}^(κ), ˜k′_(i)∈_(R)FS_(κ), ˜s_(i)∈_(R){0, 1}^(κ), and ˜s′_(i)∈_(R)FS_(κ) and computes k_(i)=tPRF(˜k_(i), ˜k′_(i), st_(i), st′_(i)) and s_(i)=tPRF(˜s_(i), ˜s′_(i), st_(i), st′_(i)) based on the twisted pseudo-random function tPRF. Further, the first key generation unit 212 computes c_(i)=g^(ki)h^(si). Then, the first key generation unit 212 transmits c_(i) to the key distribution device S.

In step S312, the key distribution device S receives (R_(i), c_(i)) or c_(i) from the communication devices U_(i). At this time, the key distribution device S stands by until the key distribution device S receives (R_(i), c_(i)), c₂, . . . , c_(n−1), (R_(n), c_(n)), . . . , (R_(n+k), c_(n+k)) respectively from all of the communication devices U₁, . . . , U_(n+k).

In step S313, the session ID generation unit 113 of the key distribution device S generates sid=H(c₁, . . . c_(n+k)) by using c₁, . . . , c_(n+k), which are received from the communication devices U₁, . . . , U_(n+k), based on the target-collision resistant hash function H. Further, one piece of communication device is selected as a representative from k+2 pieces of communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k). It is assumed that the communication device U₁ is selected, and U₁ is called a representative communication device in this example. Further, n+k−1 pieces of communication devices U_(i) (i∈{2, . . . , n+k}) other than the representative communication device U₁ are called general communication devices. In the case of i∈[n+1, n+k], the session ID generation unit 113 transmits (sid, R_(i−1), R_(i+1)) to each of the communication devices U_(i) (here, R_(n+k+1)=R₁). Further, in the case of i∈[1, 2], the session ID generation unit 113 transmits (sid, R_(i−1)) to each of the communication devices U_(i) (here, R₀=R_(n+K)). In the case of i∈[3, n−2], the session ID generation unit 113 transmits sid to each of the communication devices U_(i). In the case of i∈[n−1, n], the session ID generation unit 113 transmits (sid, R_(i+1)) to each of the communication devices U_(i). Further, the key distribution device S notifies U₁ that U₁ is the representative.

In step S413, each of the communication devices U_(i) receives any of (sid, R_(i−1), R_(i+1)), (sid, R_(i−1)), sid, and (sid, R_(i+1)) from the key distribution device S. The communication devices U_(i) execute the following processing (specifically, step S414 and step S415) as soon as the communication devices U_(i) receive any of (sid, R_(i−1), R_(i+1)), (sid, R_(i−1)), sid, and (sid, R_(i+1)). This processing is executed for six cases which are the case of i=1, the case of i=2, the case of i∈[3, n−2], the case of i=n−1, the case of i=n, and the case of i∈[n+1, n+k]. However, in the case of i∈[3, n−2], no processing is performed in step S414. That is, as soon as the communication devices U_(i) receive sid, the communication devices U_(i) execute the processing of step S415.

In the case of i=1, in step S414, the second key generation unit 214 of the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on the pseudo-random function F and generates K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function F so as to compute T₁ by an exclusive OR of K₁ ^(l) and K₁ ^(r) and compute T′ by the exclusive OR of K₁ ^(l) and k₁∥s₁, as the following formulas. Here, ∥ denotes a concatenation operator. K ₁ ^(l) =F(sid,R _(n+k) ^(r) ¹ ), K ₁ ^(r) =F(sid,g ^(r) ¹ ^(r)), T ₁ =K ₁ ^(l) ⊕K ₁ ^(r), T′=K ₁ ^(l)⊕(k ₁ ∥s ₁)

In step S415, the authentication tag generation unit 215 of the representative communication device U₁ generates an authentication tag σ₁=Tag_(mk1)(R₁, c₁, R_(n+k), T₁, T′, U₁, sid) by using the MAC key mk₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (T₁, T′, σ₁) to the key distribution device S.

In the case of i=2, in step S414, the second key generation unit 214 of the communication device U₂ generates K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function F and generates K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function F so as to compute T₂ by an exclusive OR of K₂ ^(l) and K₂ ^(r), as the following formulas. K ₂ ^(l) =F(sid,R ₁ ^(r)), K ₂ ^(r) =F(sid,g ^(r)), T ₂ =K ₂ ^(l) ⊕K ₂ ^(r)

In step S415, the authentication tag generation unit 215 of the communication device U₂ generates an authentication tag σ₂=Tag_(mk2)(c₂, R₁, k₂, s₂, T₂, U₂, sid) by using the MAC key mk_(t) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k₂, s₂, T₂, σ₂) to the key distribution device S.

In the case of i∈[3, n−2], in step S415, the authentication tag generation unit 215 of the communication devices U_(i) generates an authentication tag σ_(i)=Tag_(mki)(c_(i), k_(i), s_(i), U_(i), sid) by using the MAC key mk_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(i), s_(i), σ_(i)) to the key distribution device S.

In the case of i=n−1, in step S414, the second key generation unit 214 of the communication device U_(n−1) generates K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function F and generates K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function F so as to compute T_(n−1) by an exclusive OR of K_(n−1) ^(l) and K_(n−1) ^(r), as the following formulas. K _(n−1) ^(l) =F(sid,g ^(r)), K _(n−1) ^(r) =F(sid,R _(n) ^(r)), T _(n−1) =K _(n−) ^(l) ⊕K _(n−1) ^(r)

In step S415, the authentication tag generation unit 215 of the communication device U_(n−1) generates an authentication tag σ_(n−1)=Tag_(mkn−1)(c_(n−1), R_(n), k_(n−1), s_(n−1), T_(n−1), U_(n−1), sid) by using the MAC key mk_(n−1) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(n−1), s_(n−1), T_(n−1), σ_(n−1)) to the key distribution device S.

In the case of i=n, in step S414, the second key generation unit 214 of the communication device U_(n) generates K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function F and generates K_(n) ^(r) by using (sid, R_(n+1) ^(m)) based on the pseudo-random function F so as to compute T_(n) by an exclusive OR of K_(n) ^(l) and K_(n) ^(r), as the following formulas. K _(n) ^(l) =F(sid,R _(n) ^(r)), K _(n) ^(r) =F(sid,R _(n+1) ^(r) ^(n) ), T _(n) =K _(n) ^(l) ⊕K _(n) ^(r)

In step S415, the authentication tag generation unit 215 of the communication device U_(n) generates an authentication tag σ_(n)=Tag_(mkn)(R_(n), C_(n), R_(n+1), k_(n), s_(n), T_(n), U_(n), sid) by using the MAC key mk_(n) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(n), s_(n), T_(n), σ_(n)) to the key distribution device S.

In the case of i∈[n+1, n+k], in step S414, the second key generation unit 214 of the communication device U_(i) generates K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function F and generates K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function F so as to compute T_(i) by an exclusive OR of K_(i) ^(l) and K_(i) ^(r), as the following formulas. k _(i) ^(l) =F(sid,R _(i−1) ^(r) ^(i) ), K _(i) ^(r) =F(sid,R _(i+1) ^(r) ^(i) ), T _(i) =K _(i) ^(l) ⊕K _(i) ^(r)

In step S415, the authentication tag generation unit 215 of the communication devices U_(i) generates an authentication tag σ_(i)=Tag_(mki)(R_(i), c_(i), R_(i−1), R_(i+1), k_(i), s_(i), T_(i), U_(i), sid) by using the MAC key mk_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(i), s_(i), T_(i), σ_(i)) to the key distribution device S.

In step S314, the authentication tag verification unit 114 of the key distribution device S receives (T₁, T′, σ₁) from the representative communication devices U_(i), receives (k_(i), s_(i), T_(i), σ_(i)) from the general communication devices U_(i) with respect to i=2, n−1, . . . , n+k, and receives (k_(i), s_(i), σ_(i)) from the general communication device U_(i) with respect to i=3, . . . , n−2 so as to perform verification based on the verification algorithm Ver for a message authentication code. When the authentication tag a, (i=1, . . . , n+k) is invalid, the authentication tag verification unit 114 ends the session of the communication devices U_(i). Further, the authentication tag verification unit 114 verifies whether or not c_(i)=g^(ki)h^(si) is satisfied with respect to i=2, . . . , n+k. When c_(i)=g^(ki)h^(si) is not satisfied, the authentication tag verification unit 114 ends the session of the general communication devices U_(i).

In step S315 a, the third key generation unit 115 of the key distribution device S generates ˜k_(S)∈_(R){0, 1}^(κ) and ˜k′_(S)∈_(R)FS_(κ) so as to compute k_(S)=tPRF(˜k_(S), ˜k′_(S), st_(S), st_(S)) based on the twisted pseudo-random function tPRF. Further, the third key generation unit 115 computes k′ by the following formula. k′=(⊕_(2≤i≤n+k) k _(i))⊕k _(S)

In step S315 b, the third key generation unit 115 of the key distribution device S computes T′_(i) with respect to i=2, . . . , n+k by the following formula. T _(i)′=⊕_(1≤j≤i−1) T _(j)

Here, T_(i) is nil with respect to i=3, . . . , n−1. Accordingly, T₃′= . . . =T_(n−1)′ is obtained.

In step S315 c, the third key generation unit 115 of the key distribution device S encrypts a common key K₁ with respect to i=1, . . . , n+k based on the encryption algorithm FEnc for functional encryption with the access structure P_(i)=(ID=U_(i)){circumflex over ( )}(time∈TF) so as to generate a cipher text CT′_(i)=FEnc(Params, P_(i), K₁). Here, ID is a predicate variable representing a communication device and TF is a predicate variable representing a time frame of the communication device.

In step S316, the key distribution device S generates an authentication tag and transmits the authentication tag to the communication devices U_(i). This processing is executed for six cases which are the case of i=1, the case of i=2, the case of i∈[3, n−2], the case of i=n−1, the case of i=n, and the case of i∈[n+1, n+k].

In the case of i=1, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′₁=Tag_(mk1)(R₁, c₁, R_(n+k), T₁, T′, U₁, sid, k′, CT′₁) by using the MAC key mk₁ of the representative communication device U₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (k′, CT′_(i), σ′₁) to the representative communication device U₁.

In the case of i=2, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′₂=Tag_(mk2)(c₂, R₁, k₂, s₂, T₂, U₂, sid, c₁, k′, T′₂, T′, CT′₂) by using the MAC key mk_(t) of the general communication device U₂ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′₂, T′, CT′₂, σ′₂) to the general communication device U₂.

In the case of i∈[3, n−2], the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(i)=Tag_(mki)(c_(i), k_(i), s_(i), U_(i), sid, c₁, k′, T′_(i), T′, CT′_(i)) by using the MAC key mk_(i) of the general communication devices U_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(i), T′, CT′_(S), σ′_(i)) to the general communication device U_(i).

In the case of i=n−1, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(n−1)=Tag_(mkn−1) (c_(n−1), R_(n), k_(n−1), s_(n−1), T_(n−1), U_(n−1), sid, c₁, k′, T′_(n−1), T′, CT′_(n−1)) by using the MAC key mk_(n−1) of the general communication device U_(n−1) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(n−1), T′, CT′_(n−1), σ′_(n−1)) to the general communication device U_(n−1).

In the case of i=n, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(n)=Tag_(mkn)(R_(n), c_(n), R_(n+1), k_(n), s_(n), T_(n), U_(n), sid, c₁, k′, T′_(n), T′, CT′_(n)) by using the MAC key mk_(n) of the general communication devices U_(n) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(n), T′, CT′_(n), σ′_(n)) to the general communication device U_(n).

In the case of i∈[n+1, n+k], the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(i)=Tag_(mki)(R_(i), c_(i), R_(i−1), R_(i+1), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, CT′_(i)) by using the MAC key mk_(i) of the general communication devices U_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) to the general communication devices U_(i).

In step S416, in the case of iυ[2, n+k], the authentication tag verification unit 216 of the general communication devices U_(i) receives (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) from the key distribution device S and performs verification based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′_(i) is invalid, the authentication tag verification unit 216 ends the session of the general communication devices U_(i). Further, the authentication tag verification unit 216 computes K₁ ^(l) by an exclusive OR of T′_(i) and K_(i) ^(l) with respect to i=2, n, . . . , n+k and computes K₁ ^(l) by an exclusive OR of T′_(i) and g^(r) with respect to i=3, . . . , n−1. K ₁ ^(l) =T _(i) ′⊕K _(i) ^(l) K ₁ ^(l) =T _(i) ′⊕g ^(r)

Further, the authentication tag verification unit 216 computes k₁∥s₁ by an exclusive OR of T′ and K₁ ^(l). k ₁ ∥s ₁ =T′⊕K ₁ ^(l)

Then, the authentication tag verification unit 216 verifies whether or not c₁=g^(k1)h^(s1) is satisfied. When c_(i)=g^(k1)h^(s1) is not satisfied, the authentication tag verification unit 216 ends the session of the general communication devices U_(i).

In the case of i=1, the authentication tag verification unit 216 of the representative communication device U₁ receives (k′, CT′₁, σ′₁) from the key distribution device S and performs verification based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′₁ is invalid, the authentication tag verification unit 216 ends the session of the representative communication device U₁.

In step S417, the session key generation unit 217 of the communication devices U_(i) (i=1, . . . , n+k) decrypts the common key K₁←FDec_(uski)(CT′_(i), P_(i)) by using the user secret key usk_(i) of the communication devices U_(i) based on the decryption algorithm FDec for functional encryption. Further, the session key generation unit 217 computes a common key K₂ based on the pseudo-random function F′ as the following formula. K ₂ =F′(sid,k′⊕k ₁)

Then, the session key generation unit 217 computes a session key SK based on the pseudo-random function F″ as the following formula. SK=F″(sid,K ₁)⊕F″(sid,K ₂)

Last, the communication devices U_(i) (i=1, . . . , n) update secret information to be used for user addition and user deletion. Further, the communication devices U_(i) (i=n+1, . . . , n+k) newly store the secret information in the storage 200.

The representative communication devices U₁ update the secret information stored in the storage 200 with secret information H₁ ^(l) and r computed by the following formulas. H ₁ ^(l) =R _(n+k) ^(r) ¹ r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

The communication device U_(i) (i=2, . . . , n−1) updates the secret information stored in the storage 200 with secret information r computed by the following formula. r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

The communication devices U_(n) update the secret information stored in the storage 200 with secret information H_(n) ^(r) and r computed by the following formulas. H _(n) ^(r) =R _(n+1) ^(r) ^(n) r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

The communication device U_(i) (i=n+1, . . . , n+k) stores secret information H_(i) ^(l), H_(i) ^(r), and r computed by the following formulas in the storage 200. H _(i) ^(l) =R _(i−1) ^(r) ^(i) H _(i) ^(r) =R _(i+1) ^(r) ^(i) r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

According to the key exchange technology of the present invention, the above-described configuration makes it unnecessary to preliminarily register information of users who perform key exchange as the related art, thus enabling efficient share of common key at the time of user addition as well. Specifically, the whole computational complexity required for the key exchange is the constant number of times which is the number of users, that is, O(1).

Here, the processing for guaranteeing integrity by using the message authentication code algorithm, that is, the processing for generation and verification of an authentication tag in S415, S314, S316, and S416 may be omitted.

Further, the communication devices U_(i) are configured to generate two keys, that is, the common key K₂ and the session key SK from sid based on the pseudo-random function so as to share these two keys among the communication devices U_(i). However, the communication devices U_(i) may be configured to generate and share only the common key K₂.

<User Deletion>

A processing procedure of user deletion in the key exchange method according to the embodiment will be described with reference to FIG. 6.

It is assumed that communication devices U_(j1), . . . , U_(jm) leave from a session established by the communication devices U₁, . . . , U_(n) (here, m is an integer which is 1 or larger). Hereinafter, it is assumed that R={U_(j1), . . . , U_(jm)} holds. Further, it is assumed that N={U_(j1−1), U_(j1+1), . . . , U_(jm−1), U_(jm+1)} holds. Accordingly, both of R and N are subsets of {U₁, . . . , U_(n)}.

Here, even in a case of U₁∈N, generality is not lost. Therefore, U₁∈N is assumed.

In step S511, in the case where a session is started by the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) and the session is the first session in a time frame TF′ of the communication devices U the user key transmission unit 111 of the key distribution device S generates a user secret key usk_(i)←Der(Params, msk, A_(i)) of the communication devices U_(i) based on the key derivation algorithm Der for functional encryption, with current time and attribution respectively used as time and A_(i)=(U_(i), time). Further, the user key transmission unit 111 generates a MAC key mk_(i)←MGen of the communication devices U_(i) based on the key generation algorithm MGen for a message authentication code. Then, the user key transmission unit 111 encrypts the user secret key usk_(i) and the MAC key mk_(i) by using the public key pk_(i) of the communication device U_(i) based on the encryption algorithm Enc for public key encryption so as to generate the cipher text CT_(i)←Enc_(pki)(usk_(i), mk_(i)). The user key transmission unit 111 transmits the cipher text CT_(i) to each of the communication devices U_(i).

Further, in the case where a time frame has been changed from the session established by the communication device U₁, . . . , U_(n), that is, in the case where TF′ and TF are not equal to each other, the key distribution device S generates ˜K₁∈_(R) {0, 1}^(κ) and ˜K′₁∈_(R)FS_(κ) so as to compute K₁=tPRF′(˜K₁, ˜K′₁, st_(S), st′_(S)) based on the twisted pseudo-random function tPRF′. Here, the computation of K₁ is performed by the third key generation unit 115 of the key distribution device S in step S515 a which will be described later, so that the computation does not have to be performed here.

In step S611, the user key reception unit 211 of the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) decrypts the cipher text CT_(i), which is received from the key distribution device S, by using the secret key sk_(i) of the communication devices U_(i) based on the decryption algorithm Dec for public key encryption so as to obtain a user secret key and a MAC key (usk_(i), mk_(i))←Dec_(ski)(CT_(i)). The user key reception unit 211 stores the user secret key usk_(i) and the MAC key mk_(i) in the storage 200.

In step S612, in the case of U_(i)∈N, the first key generation unit 212 of the communication devices U_(i) generates ˜r_(i)∈_(R){0, 1}^(κ), ˜r′_(i)∈_(R)FS_(κ), ˜k_(i)∈_(R){0, 1}^(κ), ˜k′_(i)∈_(R)FS_(κ), ˜s_(i)∈_(R){0, 1}^(κ), and ˜s′_(i)∈_(R)FS_(κ) and computes r_(i)=tPRF(˜r_(i), ˜r′_(i), st_(i), st′_(i)), k_(i)=tPRF(˜k_(i), ˜k′_(i), st_(i), st′_(i)), and s_(i)=tPRF(˜s_(i), ˜s′_(i), st_(i), st′_(i)) based on the twisted pseudo-random function tPRF. Further, the first key generation unit 212 computes R_(i)=g^(ri) and c_(i)=g^(ki)h^(si). Then, the first key generation unit 212 transmits (R_(i), c_(i)) to the key distribution device S.

In the case of U_(i)∈({U₁, . . . , U_(n)}−R)−N, the first key generation unit 212 of the communication devices U_(i) generates ˜k_(i)∈_(R){0, 1}^(κ), ˜k′_(i)∈_(R)FS_(κ), ˜s_(i)∈_(R){0, 1}^(κ), and ˜s′_(i)∈_(R)FS_(κ) and computes k_(i)=tPRF(˜k_(i), ˜k′_(i), st_(i), st′_(i)) and s_(i)=tPRF(˜s_(i), ˜s′_(i), st_(i), st′_(i)) based on the twisted pseudo-random function tPRF. Further, the first key generation unit 212 computes c_(i)=g^(ki)h^(si). Then, the first key generation unit 212 transmits c_(i) to the key distribution device S.

In step S512, the key distribution device S receives (R_(i), c_(i)) from the communication devices U_(i)∈N and receives c_(i) from the communication devices U_(i)∈({U_(i), U_(n)}−R)−N. At this time, the key distribution device S stands by until the key distribution device S receives (R_(i), c_(i)) or c_(i) from all of the communication devices U_(i).

In step S513, the session ID generation unit 113 of the key distribution device S generates sid=H({c_(i)}) by using {c_(i)}, which are received from the communication devices U_(i)∈{U₁, . . . , U_(n)}−R, based on the target-collision resistant hash function H. Further, one piece of communication device is selected as a representative from the communication devices U_(i)∈N. It is assumed that the communication device U₁ is selected, and U₁ is called a representative communication device in this example. Further, n−1−m pieces of communication devices U_(i)∈{U₂, . . . , U_(n)}−R other than the representative communication device U₁ are called general communication devices. With respect to i which satisfies U_(i)∈N and U_(i+1)∈R, the session ID generation unit 113 transmits (sid, R_(j)) to each of the communication devices U_(i) (here, j is a minimum index which satisfies U_(j)∈N and j>i). Further, with respect to i which satisfies U_(i)∈N and U_(i−1)∈R, the session ID generation unit 113 transmits (sid, R_(j′)) to each of the communication devices U_(i) (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i). With respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, the session ID generation unit 113 transmits sid to each of the communication devices U_(i). Further, the key distribution device S notifies U₁ that U₁ is the representative.

In step S613, each of the communication devices U_(i) receives any of (sid, R_(j)), (sid, R_(j′)), and sid from the key distribution device S. The communication devices U_(i) execute the following processing (specifically, step S614 and step S615) as soon as the communication devices U_(i) receive any of (sid, R_(j)), (sid, R_(j′)), and sid. This processing is executed for four cases which are the case of i=1, the case of i which satisfies U_(i)∈N and U_(i+1)∈R (i is not 1), the case of i which satisfies U_(i)∈N and U_(i−1)∈R (i is not 1), and the case of i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (i is not 1). Further, in the case of i=1, this processing is executed for three cases.

In the case of i=1, in step S614, any of a couple of U_(j)=U₃ and U_(j′)=U_(n−1), a couple of U_(j′)=U_(n−1) and U₂∈N, and a couple of U_(j)=U₃ and U_(n)∈N is satisfied. When the couple of U_(j)=U₃ and U_(j′)=U_(n−1) is satisfied, the second key generation unit 214 of the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function F and generates K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function F so as to compute T₁ by the exclusive OR of K₁ ^(l) and K₁ ^(r) and compute T′ by the exclusive OR of K₁ ^(l) and k₁∥s₁, as the following formulas. Here, ∥ denotes a concatenation operator. K ₁ ^(l) =F(sid R _(n−1) ^(r) ¹ ), K ₁ ^(r) =F(sid,R ₃ ^(r) ¹ ), T ₁ =K ₁ ^(l) ⊕K ₁ ^(r), T′=K ₁ ^(l)⊕(k ₁ ∥s ₁)

When the couple of U_(j′)=U_(n−1) and U₂∈N is satisfied, the second key generation unit 214 of the representative communication device U₁ generates K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function F and generates K₁ ^(r) by using (sid, H₁ ^(r)) based on the pseudo-random function F so as to compute T₁ by the exclusive OR of K₁ ^(l) and K₁ ^(r) and compute T′ by the exclusive OR of K₁ ^(l) and k₁∥s₁, as the following formulas. Here, ∥ denotes a concatenation operator. K ₁ ^(l) =F(sid,R _(n−1) ^(r) ¹ ), K ₁ ^(r) =F(sid,H ₁ ^(r)), T ₁ =K ₁ ^(l) ⊕K ₁ ^(r), T′=K ₁ ^(l)⊕(k ₁ ∥s ₁)

When the couple of U_(j)=U₃ and U_(n)∈N is satisfied, the second key generation unit 214 of the representative communication device U₁ generates K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function F and generates K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function F so as to compute T₁ by the exclusive OR of K₁ ^(l) and K₁ ^(r) and compute T′ by the exclusive OR of K₁ ^(l) and k₁∥s_(i), as the following formulas. Here, ∥ denotes a concatenation operator. K ₁ ^(l) =F(sid,H ₁ ^(l)), K ₁ ^(r) =F(sid,R ₃ ^(r) ¹ ), T ₁ =K ₁ ^(l) ⊕K ₁ ^(r), T′=K ₁ ^(l)(k ₁ ∥s ₁)

In step S615, the authentication tag generation unit 215 of the representative communication device U₁ generates an authentication tag σ₁=Tag_(mk1)(R₁, c₁, (R₃, R_(n−1)), T₁, T′, U₁, sid) by using the MAC key mk₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (T₁, T′, σ₁) to the key distribution device S.

With respect to i which satisfies the couple of U_(i)∈N and U_(i+1)∈R (i is not 1), in step S614, the second key generation unit 214 of the communication devices U_(i) generates K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function F and generates K_(i) ^(r) by using (sid, R_(j) ^(ri)) based on the pseudo-random function F so as to compute T_(i) by the exclusive OR of K_(i) ^(l) and K_(i) ^(r), as the following formulas. K _(i) ^(l) =F(sid,H _(i) ^(l)), K _(i) ^(r) =F(sid,R _(j) ^(r) ^(i) ), T _(i) =K _(i) ^(l) ⊕K _(i) ^(r)

In step S615, the authentication tag generation unit 215 of the communication devices U_(i) generates an authentication tag σ_(i)=Tag_(mki)(R_(i), c_(i), R_(j), k_(i), s_(i), T_(i), U_(i), sid) by using the MAC key mk_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(i), s_(i), T_(i), σ_(j)) to the key distribution device S.

With respect to i which satisfies the couple of U_(i)∈N and U_(i−1)∈R (i is not 1), in step S614, the second key generation unit 214 of the communication devices U_(i) generates K_(i) ^(l) by using (sid, R_(j′) ^(ri)) based on the pseudo-random function F and generates K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function F so as to compute T_(i) by the exclusive OR of K_(i) ^(l) and K_(i) ^(r), as the following formulas. K _(i) ^(l) =F(sid,R _(j) ^(r) ^(i) ), K _(i) ^(r) =F(sid,H _(i) ^(r)), T _(i) =K _(i) ^(l) ⊕K _(i) ^(r)

In step S615, the authentication tag generation unit 215 of the communication devices U_(i) generates an authentication tag σ_(i)=Tag_(mki)(R_(i), c_(i), R_(j′), k_(i), s_(i), T_(i), U_(i), sid) by using the MAC key mk_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(i), s_(i), T_(i), σ_(i)) to the key distribution device S.

With respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (i is not 1), in step S614, the second key generation unit 214 of the communication device U_(i) generates K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function F and generates K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function F so as to compute T_(i) by the exclusive OR of K_(i) ^(l) and K_(i) ^(r), as the following formulas. K _(i) ^(l) =F(sid,H _(i) ^(l)), K _(i) ^(r) =F(sid,H _(i) ^(r)), T _(i) =K _(i) ^(l) ⊕K _(i) ^(r)

In step S615, the authentication tag generation unit 215 of the communication devices U_(i) generates an authentication tag σ_(i)=Tag_(mki)(c_(i), k_(i), s_(i), T_(i), U_(i), sid) by using the MAC key mk_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 215 transmits (k_(i), s_(i), T_(i), σ_(i)) to the key distribution device S.

In step S514, the authentication tag verification unit 114 of the key distribution device S receives (T₁, T′, σ₁) from the representative communication device U₁, and receives (k_(i), s_(i), T_(i), σ_(i)) from the general communication devices U_(i) other than the representative communication device U₁, that is, from U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁} so as to perform verification based on the verification algorithm Ver for a message authentication code. When the authentication tag σ_(i) is invalid, the authentication tag verification unit 114 ends the session of the communication devices U_(i). The authentication tag verification unit 114 verifies whether or not c_(i)=g^(ki)h^(si) is satisfied in the case of U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}. When c_(i)=g^(ki)h^(si) is not satisfied, the authentication tag verification unit 114 ends the session of the general communication devices U_(i).

In step S515 a, the third key generation unit 115 of the key distribution device S generates ˜k_(S)∈_(R){0, 1}^(κ) and ˜k′_(S)∈_(R)FS_(κ) so as to compute k_(S)=tPRF(˜k_(S), ˜k′_(S), st_(S), st′_(S)) based on the twisted pseudo-random function tPRF. The third key generation unit 115 computes k′ by the following formula with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}. k′=(⊕{k _(i)})⊕k _(S)

In step S515 b, the third key generation unit 115 of the key distribution device S computes T′_(i) by the following formula in the case of U_(i)∈{U₁, . . . , U_(n)}−R. T _(i′)=⊕_(1≤j≤i−1) Tj

Here, T_(j) is nil with respect to j which satisfies U_(j)∈R.

In step S515 c, the third key generation unit 115 of the key distribution device S encrypts the common key K₁ with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R based on the encryption algorithm FEnc for functional encryption with the access structure P_(i)=(ID=U_(i)){circumflex over ( )}(time∈TF) so as to generate a cipher text CT′_(i)=FEnc(Params, P_(i), K₁). Here, ID is a predicate variable representing a communication device and TF is a predicate variable representing a time frame of the communication device.

In step S516, the key distribution device S generates an authentication tag and transmits the authentication tag to the communication devices Ui. This processing is executed for four cases which are the case of i=1, the case of i which satisfies U_(i)∈N and U_(i+1)∈R, the case of i which satisfies U_(i)∈N and U_(i−1)∈R, and the case of i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N.

In the case of i=1, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′₁=Tag_(mk1)(R₁, c₁, (R₃, R_(n−1)), T₁, T′, U₁, sid, k′, CT′₁) by using the MAC key mk₁ of the representative communication device U₁ based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (k′, CT′₁, σ′₁) to the representative communication device U₁.

In the case of i which satisfies U_(i)∈N and U_(i+1)∈R, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(i)=Tag_(mki)(R_(i), c_(i), R_(j), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′₁, CT′_(i)) by using the MAC key mk_(i) of the general communication devices U_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) to the general communication devices U_(i).

In the case of i which satisfies U_(i)∈N and U_(i−1)∈R, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(i)=Tag_(mki)(R_(i), c_(i), R_(j′), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, CT′_(i)) by using the MAC key mk_(i) of the general communication devices U_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c_(i), k′, T′_(i), T′, CT′_(i), σ′_(i)) to the general communication devices U_(i).

In the case of i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, the authentication tag generation unit 116 of the key distribution device S generates an authentication tag σ′_(i)=Tag_(mki)(c_(i), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, CT′_(i)) by using the MAC key mk_(i) of the general communication devices U_(i) based on the tag generation algorithm Tag for a message authentication code. The authentication tag generation unit 116 transmits (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) to the general communication device U_(i).

In step S616, with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}, the authentication tag verification unit 216 of the general communication devices U_(i) receives (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) from the key distribution device S and performs verification based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′_(i) is invalid, the authentication tag verification unit 216 ends the session of the general communication devices U_(i). Further, the authentication tag verification unit 216 computes K₁ ^(l) by the exclusive OR of T′_(i) and K_(i) ^(l) and computes k₁∥s₁ by the exclusive OR of T′ and K₁ ^(l), as the following formulas. K ₁ ^(l) =T _(i) ′⊕K _(i) ^(l) k ₁ ∥s ₁ =T′⊕K ₁ ^(l)

Then, the authentication tag verification unit 216 verifies whether or not c₁=g^(k1)h^(s1) is satisfied. When c₁=g^(k1)h^(s1) is not satisfied, the authentication tag verification unit 216 ends the session of the general communication devices U_(i).

In the case of i=1, the authentication tag verification unit 216 of the representative communication device U₁ receives (k′, CT′₁, σ′₁) from the key distribution device S and performs verification based on the verification algorithm Ver for a message authentication code. When the authentication tag σ′₁ is invalid, the authentication tag verification unit 216 ends the session of the representative communication device U₁.

In step S617, the session key generation unit 217 of the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) decrypts the common key K₁∂FIDec_(uski)(CT′_(i), P_(i)) by using the user secret key usk_(i) of the communication devices U_(i) based on the decryption algorithm FDec for functional encryption. Further, the session key generation unit 217 computes the common key K₂ based on the pseudo-random function F′ as the following formula. K ₂ =F′(sid,k′⊕k ₁)

Then, the session key generation unit 217 computes the session key SK based on the pseudo-random function F″ as the following formula. SK=F″(sid,K ₁)⊕F″(sid,K ₂)

Last, the communication devices U_(i)∈{U₁, . . . , U_(n)}−R update secret information to be used for user addition and user deletion.

The communication devices U_(i)(∈{U₁, . . . , U_(n)}−R) update the secret information stored in the storage 200 with secret information r computed by the following formula. r=F′″(sid,K ₁)⊕F′″(sid,K ₂)

Further, the representative communication device U_(i) updates the secret information stored in the storage 200 with secret information H₁ ^(r) (the case of U₂∈R) and H₁ ^(l) (the case of U_(n)∈R) respectively computed by the following formulas. H ₁ ^(r) =R _(j) ^(r) ¹ H ₁ ^(l) =R _(j′) ^(r) ¹

With respect to i which satisfies U_(i)∈N and U_(i+1)∈R, the communication devices U_(i) update the secret information stored in the storage 200 with secret information H_(i) ^(r) computed by the following formula. H _(i) ^(r) =R _(j) ^(r) ^(i)

With respect to i which satisfies U_(i)∈N and U_(i−1)∈R, the communication devices U_(i) update the secret information stored in the storage 200 with secret information H_(i) ^(l) computed by the following formula. H _(i) ^(l) =R _(j′) ^(r) ^(i)

According to the key exchange technology of the present invention, the above-described configuration makes it unnecessary to preliminarily register information of users who perform key exchange as the related art, thus enabling efficient share of common key at the time of user deletion as well. Specifically, the whole computational complexity required for the key exchange is the constant number of times which is the number of users, that is, O(1).

Here, the processing for guaranteeing integrity by using the message authentication code algorithm, that is, the processing for generation and verification of an authentication tag in S615, S514, S516, and S616 may be omitted.

Further, the communication devices U_(i) are configured to generate two keys, that is, the common key K₂ and the session key SK from sid based on the pseudo-random function so as to share these two keys among the communication devices U_(i). However, the communication devices U_(i) may be configured to generate and share only the common key K₂.

It is obvious that the present invention is not limited to the above-described embodiment and alterations can be made as appropriate within a scope of the idea of the present invention. Various types of processing which are described in the above embodiment may be executed in time series in accordance with the described order and may be executed in parallel or individually in accordance with the processing capacity of the device performing the processing or in accordance with the need.

Furthermore, the above-described embodiment is presented for the purpose of exemplification and description. There is no encompassing intention and further, there is no intention to precisely limit the invention to the disclosed form. Modifications and variations can be made based on the above-described instruction. The embodiment is selected and presented so as to provide the best exemplification of the principle of the present invention and to enable those skilled in the art to use the present invention in various embodiments and use the present invention by adding various modifications for adaptation of the present invention to well-thought actual use. All of such modifications and variations are within the scope of the present invention defined by the added claims interpreted in accordance with the justly-, legally-, and fairly-granted latitude.

[Program and Recording Medium]

When various types of processing functions in the devices described in the above embodiment are implemented on a computer, the contents of processing function to be contained in each device is written by a program. With this program executed on the computer, various types of processing functions in the above-described devices are implemented on the computer.

This program in which the contents of processing are written can be recorded in a computer-readable recording medium. The computer-readable recording medium may be any medium such as a magnetic recording device, an optical disc, a magneto-optical recording medium, and a semiconductor memory.

Distribution of this program is implemented by sales, transfer, rental, and other transactions of a portable recording medium such as a DVD and a CD-ROM on which the program is recorded, for example. Furthermore, this program may be stored in a storage unit of a server computer and transferred from the server computer to other computers via a network so as to be distributed.

A computer which executes such program first stores the program recorded in a portable recording medium or transferred from a server computer once in a storage unit of the computer, for example. When the processing is performed, the computer reads out the program stored in the recording medium of the computer and performs processing in accordance with the program thus read out. As another execution form of this program, the computer may directly read out the program from a portable recording medium and perform processing in accordance with the program. Furthermore, each time the program is transferred to the computer from the server computer, the computer may sequentially perform processing in accordance with the received program. Alternatively, a configuration may be adopted in which the transfer of a program to the computer from the server computer is not performed and the above-described processing is executed by so-called application service provider (ASP)-type service by which the processing functions are implemented only by an instruction for execution thereof and result acquisition. It should be noted that a program according to the present embodiment includes information which is provided for processing performed by electronic calculation equipment and which is equivalent to a program (such as data which is not a direct instruction to the computer but has a property specifying the processing performed by the computer).

In the present embodiment, the present device is configured with a predetermined program executed on a computer. However, the present device may be configured with at least part of these processing contents realized in a hardware manner.

The foregoing description of the embodiment of the invention has been presented for the purpose of illustration and description. It is not intended to be exhaustive and to limit the invention to the precise form disclosed. Modifications or variations are possible in light of the above teaching. The embodiment was chosen and described to provide the best illustration of the principles of the invention and its practical application, and to enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled. 

What is claimed is:
 1. A key exchange method for a case where, in a key exchange system which includes a key distribution device S and n+k pieces (here, n is an integer which is 2 or larger and k is an integer which is 1 or larger) of communication devices U_(i) (i=1, . . . , n+k), communication devices U_(n+1), . . . , U_(n+k) newly join a session established by communication devices U₁, U_(n), in which ∥ is a concatenation operator, ₁ is one piece of representative communication device which is selected from the communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k), a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S are stored in a storage of the key distribution device S, a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) are stored in a storage of the communication devices U_(i) (i=1, . . . , n+k), and further, information r generated in the session established by the communication devices U₁, . . . , U_(n) is stored in a storage of the communication devices U₁, . . . , U_(n), the key exchange method comprising: a first key generation step in which the communication devices U_(i) (i=1, n, . . . , n+k) generate values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, compute values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmit (R_(i), c_(i)) to the key distribution device S, and the communication devices U_(i) (i=2, . . . , n−1) generate values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, compute a value c_(i)=g^(ki)h^(si), and transmit the value c_(i) to the key distribution device S; a session ID generation step in which the key distribution device S generates a value sid by using the values c₁, . . . , c_(n+k) based on a target-collision resistant hash function and transmits, to the communication devices U_(i), (sid, R_(i−1)) with respect to i=1, 2, the value sid with respect to i=3, . . . , n−2, (sid, R_(i+1)) with respect to i=n−1, n, and (sid, R_(i−1), R_(i+1)) with respect to i=n+1, . . . , n+k (here, R₀=R_(n+K) and R_(n+k+1)=R₁); a second key generation step in which the representative communication device U₁ generates a value K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on a pseudo-random function, generates a value K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function, computes a value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes a value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S, the communication device U₂ generates a value K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function, generates a value K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function, computes a value T₂ by an exclusive OR of the values K₂ ^(l) and K₂ ^(r), and transmits (k₂, s₂, T₂) to the key distribution device S, the communication devices U (i=3, . . . , n−2) transmit (k_(i), s_(i)) to the key distribution device S, the communication device U_(n−1) generates a value K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function, generates a value K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function, computes a value T_(n−1) by an exclusive OR of the values K_(n−1) ^(l) and K_(n−1) ^(r), and transmits (k_(n−1), s_(n−1), T_(n−1)) to the key distribution device S, the communication device U_(n) generates a value K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function, generates a value K_(n) ^(r) by using (sid, R_(n+1) ^(rn)) based on the pseudo-random function, computes a value T_(n) by an exclusive OR of the values K_(n) ^(l) and K_(n) ^(r), and transmits (k_(n), s_(n), T_(n)) to the key distribution device S, and the communication devices U_(i) (i=n+1, n+k) generate a value K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function, generate a value K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function, compute a value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S; a third key generation step in which the key distribution device S generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes a value k′ by an exclusive OR of the values k₂, . . . , k_(n+k), k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(i−1) with respect to i=2, . . . , n+k (here, T_(i) is nil with respect to i=3, . . . , n−1), transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (i=2, . . . , n+k); a first session key generation step in which the communication devices U_(i) (i=2, n, . . . , n+k) compute the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l) and compute k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and the communication devices U_(i) (i=3, . . . , n−1) compute the value K₁ ^(l) by an exclusive OR of the values T′_(i) and g^(r) and compute k₁∥s₁ by the exclusive OR of the values T′ and K₁ ^(l); and a second session key generation step in which the communication devices U_(i) (i=1, . . . , n+k) generate a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function.
 2. The key exchange method according to claim 1, in which time denotes current time, ID denotes a predicate variable representing the communication device, and TF denotes a predicate variable representing a time frame of the communication device, and a master secret key msk for functional encryption and a common key K₁ which is generated in a session established by the communication devices U₁, . . . , U_(n) are further stored in the storage of the key distribution device S, the key exchange method further comprising: a user key transmission step in which the key distribution device S generates a user secret key usk_(i) by using the master secret key msk with respect to i=1, . . . , n+k with attribution used as A_(i)=(U_(i), time) based on a key derivation algorithm for functional encryption, and encrypts the user secret key usk_(i) by using a public key pk_(i) of the communication devices U_(i) based on an encryption algorithm for public key encryption so as to generate a cipher text CT_(i); and a user key reception step in which the communication devices U_(i) (i=1, . . . , n+k) decrypt the cipher text CT_(i) by using the secret key sk_(i) based on a decryption algorithm for public key encryption so as to obtain the user secret key usk_(i), wherein in the third key generation step, in a case where a time frame has been changed from the session established by the communication devices U₁, . . . , U_(n), the key distribution device S generates the common key K₁ by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, and in a case where a time frame has not been changed from the session established by the communication devices U₁, . . . , U_(n), the key distribution device S reads the common key K₁ stored in the storage and encrypts the common key K₁ with respect to i=1, . . . , n+k with an access structure P_(i)=(ID=U_(i))∧(time∈TF) based on an encryption algorithm for functional encryption so as to further generate a cipher text CT′_(i), and in the second session key generation step, the communication devices U_(i) (i=1, . . . , n+k) decrypt the cipher text CT′_(i) by using the user secret key usk_(i) based on a decryption algorithm for functional encryption so as to obtain the common key K₁ and further compute a session key SK by an exclusive OR of a value generated by using (sid, K₁) based on the pseudo-random function and a value generated by using (sid, K₂) based on the pseudo-random function.
 3. The key exchange method according to claim 2, in which in the user key transmission step, the key distribution device S further generates a MAC key mk_(i) based on a key generation algorithm for a message authentication code and encrypts the user secret key usk_(i) and the MAC key mk_(i) by using the public key pk_(i) of the communication devices U_(i) based on the encryption algorithm for public key encryption so as to generate the cipher text CT_(i), and in the user key reception step, the communication devices U_(i) (i=1, . . . , n+k) decrypt the cipher text CT_(i) by using the secret key sk_(i) based on the decryption algorithm for public key encryption so as to obtain the user secret key usk_(i) and the MAC key mk_(i), the key exchange method further comprising: a first authentication tag generation step in which the representative communication device U₁ generates an authentication tag σ₁ by using the MAC key mk₁ and the values R₁, c₁, R_(n+k), T₁, T′, U₁, and sid based on a tag generation algorithm for a message authentication code, the communication device U₂ generates an authentication tag σ₂ by using the MAC key mk₂ and the values c₂, R₁, k₂, s₂, T₂, U₂, and sid based on the tag generation algorithm for a message authentication code, the communication devices U_(i) (i=3, . . . , n−2) generate an authentication tag σ_(i) by using the MAC key mk_(i) and the values c_(i), k_(i), s_(i), U_(i), and sid based on the tag generation algorithm for a message authentication code, the communication device U_(n−1) generates an authentication tag σ_(n−1) by using the MAC key mk_(n−1) and the values c_(n−1), R_(n), k_(n−1), s_(n−1), T_(n−1), U_(n−1), and sid based on the tag generation algorithm for a message authentication code, the communication device U_(n) generates an authentication tag σ_(n) by using the MAC key mk_(n) and the values R_(n), c_(n), R_(n+1), k_(n), s_(n), T_(n), U_(n), and sid based on the tag generation algorithm for a message authentication code, and the communication devices U_(i) (i=n+1, . . . , n+k) generate an authentication tag σ_(i) by using the MAC key mk_(i) and the values R_(i), c_(i), R_(i−1), R_(i+1), k_(i), s_(i), T_(i), U_(i), and sid based on the tag generation algorithm for a message authentication code; a first authentication tag verification step in which the key distribution device S receives (T₁, T′, σ₁) from the representative communication device U₁, receives (k_(i), s_(i), T_(i), σ_(i)) from the communication devices U_(i) (i=2, n−1, . . . , n+k), receives (k_(i), s_(i), σ_(i)) from the communication devices U_(i) (i=3, . . . , n−2), verifies the authentication tag σ_(i) by using the MAC key mk_(i) (i=1, . . . , n+k) based on a verification algorithm for a message authentication code, and verifies whether or not c_(i)=g^(ki)h^(si) is satisfied with respect to i=2, . . . , n+k; a second authentication tag generation step in which the key distribution device S generates an authentication tag σ′₁ with respect to i=1 by using the MAC key mk₁ and the values R₁, c₁, R_(n+k), T₁, T′, U₁, sid, k′, and CT′₁ based on the tag generation algorithm for a message authentication code, generates an authentication tag σ′₂ with respect to i=2 by using the MAC key mk₂ and the values c₂, R₁, k₂, s₂, T₂, U₂, sid, c₁, k′, T′₂, T′, and CT′₂ based on the tag generation algorithm for a message authentication code, generates an authentication tag with respect to i=3, . . . , n−2 by using the MAC key mk_(i) and the values c_(i), k_(i), s_(i), U_(i), sid, c₁, k′, T′_(i), T′, and CT′_(i) based on the tag generation algorithm for a message authentication code, generates an authentication tag σ′_(n−1) with respect to i=n−1 by using the MAC key mk_(n−1) and the values c_(n−1), R_(n), k_(n−1), s_(n−1), T_(n−1), U_(n−1), sid, c₁, k′, T′_(n−1), T′, and CT′_(n−1) based on the tag generation algorithm for a message authentication code, generates an authentication tag σ′_(n) with respect to i=n by using the MAC key mk_(n) and the values R_(n), c_(n), R_(n+1), k_(n), s_(n), T_(n), U_(n), sid, c₁, k′, T′_(n), T′, and CT′_(n) based on the tag generation algorithm for a message authentication code, and generates an authentication tag σ′_(i) with respect to i=n+1, . . . , n+k by using the MAC key mk_(i) and the values R_(i), c_(i), R_(i−1), R_(i+1), k_(i), s_(i), T_(i), sid, c₁, k′, T′_(i), and CT′_(i) based on the tag generation algorithm for a message authentication code; and a second authentication tag verification step in which the representative communication device U₁ receives (k′, CT′₁, σ′₁) from the key distribution device S and verifies the authentication tag σ′₁ by using the MAC key mk₁ based on the verification algorithm for a message authentication code, and the communication devices U_(i) (i=2, . . . , n+k) receive (c₁, k′, T′_(i), T′, CT′_(i), σ′_(i)) from the key distribution device S, verify the authentication tag σ′_(i), by using the MAC key mk_(i) based on the verification algorithm for a message authentication code, and verify whether or not c_(i)=g^(k1)h^(s1) is satisfied.
 4. A key exchange system comprising: a key distribution device S; and n+k pieces (here, n is an integer which is 2 or larger and k is an integer which is 1 or larger) of communication devices U_(i) (i=1, . . . , n+k), wherein ∥ is a concatenation operator, U₁ is one piece of representative communication device which is selected from the communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k), the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i), c_(i)) from the communication devices U_(i) (i=1, . . . n, n+k), receives a value c_(i) from the communication devices U₁ (i=2, . . . , n−1), generates a value sid by using the values c₁, . . . , c_(n+k) based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(i−1)) with respect to i=1, 2, the value sid with respect to i=3, . . . n−2, (sid, R_(i+1)) with respect to i=n−1, n, and (sid, R_(i−1), R_(i+1)) with respect to i=n+1, . . . , n+k (here, R₀=R_(n+K), R_(n+k+1)=R₁), and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication device U_(i) (i=2, n−1, . . . , n+k), receives (k_(i), s_(i)) from the communication devices U_(i) (i=3, . . . , n−2), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on a twisted pseudo-random function, computes a value k′ by an exclusive OR of the values k₂, . . . , k_(n+k), k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(i−1) with respect to i=2, . . . , n+k (here, T_(i) is nil with respect to i=3, . . . , n−1), transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (i=2, . . . , n+k), and the communication devices U_(i) (i=1, . . . , n+k) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and further stores information r generated in a session established by the communication devices U₁, . . . , U_(n) with respect to i=1, . . . , n, circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i=1, n, . . . , n+k, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i=2, . . . , n−1, execute a second key generation processing which receives (sid, R_(n+k)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S with respect to i=1, receives (sid, R₁) from the key distribution device S, generates a value K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function, generates a value K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function, computes the value T₂ by an exclusive OR of the values K₂ ^(l) and K₂ ^(r), and transmits (k₂, s₂, T₂) to the key distribution device S with respect to i=2, receives the value sid from the key distribution device S and transmits (k_(i), s_(i)) to the key distribution device S with respect to i=3, . . . , n−2, receives (sid, R_(n)) from the key distribution device S, generates a value K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function, generates a value K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function, computes the value T_(n−1) by an exclusive OR of the values K_(n−1) ^(l) and K_(n−1) ^(r), and transmits (k_(n−1), s_(n−1), T_(n−1)) to the key distribution device S with respect to i=n−1, receives (sid, R_(n+1)) from the key distribution device S, generates a value K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function, generates a value K_(n) ^(r) by using (sid, R_(n+1) ^(rn)) based on the pseudo-random function, computes the value T_(n) by an exclusive OR of the values K_(n) ^(l) and K_(n) ^(r), and transmits (k_(n), s_(n), T_(n)) to the key distribution device S with respect to i=n, and receives (sid, R_(i−1), R_(i+1)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S with respect to i=n+1, . . . , n+k, and execute a session key generation processing which receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=1, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=2, n, . . . , n+k, and receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and g^(r), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=3, . . . , n−1.
 5. A key exchange method for a case where, in a key exchange system which includes a key distribution device S and n pieces (here, n is an integer which is 2 or larger) of communication devices U_(i) (i=1, . . . , n), communication devices U_(j1), . . . , U_(jm) leave from a session established by communication devices U₁, . . . , U_(n), in which R={U_(j1), . . . , U_(jm)} is a subset of {U₁, . . . , U_(n)} and N={U_(j1-1), U_(j1+1), . . . , U_(jm−1), U_(jm+1)} is a subset of {U₁, . . . , U_(n)}, ∥ is a concatenation operator, U₁ (∈N) is one piece of representative communication device which is selected from N, a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S are stored in a storage of the key distribution device S, and a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and information H_(i) ^(l) and H_(i) ^(r) generated in the session established by the communication devices U₁, . . . , U_(n) are stored in a storage of the communication devices U_(i) (i=1, . . . , n), the key exchange method comprising: a first key generation step in which the communication devices U_(i) (∈N) generate values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, compute values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmit (R_(i), c_(i)) to the key distribution device S, and the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−N) generate values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, compute a value c_(i)=g^(ki)h^(si), and transmit the value c_(i) to the key distribution device S; a session ID generation step in which the key distribution device S generates a value sid by using {c_(i)|i satisfies U_(i)∈{U₁, . . . U_(n)}−R} based on a target-collision resistant hash function and transmits, to the communication devices U_(i), (sid, R_(j)) with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (here, j is a minimum index which satisfies U_(j)∈N and j>i), (sid, R_(j′)) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i), and the value sid with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N; a second key generations step in which the representative communication device U₁ generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on a pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes a value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes a value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(j′)=U_(n−1) are satisfied, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₁ ^(r)) based on the pseudo-random function, computes a value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes a value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j′)=U_(n−1) and U₂∈N are satisfied, and generates a value K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes a value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes a value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(n)∈N are satisfied, the communication devices U_(i) (i satisfies U_(i)∈N and U_(i+1)∈R (here, i is not 1)) generate a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generate a value K_(i) ^(r) by using (sid, R_(j) ^(ri)) based on the pseudo-random function, compute a value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S, the communication devices U_(i) (i satisfies U_(i)∈N and U_(i−1)∈R (here, i is not 1)) generate a value by using (sid, R_(j′) ^(ri)) based on the pseudo-random function, generate a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, compute a value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S, and the communication devices U_(i) (i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (here, i is not 1)) generate a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generate a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, compute a value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmit (k_(i), s_(i), T_(i)) to the key distribution device S; a third key generation step in which the key distribution device S generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes a value k′ by an exclusive OR of {k_(i)|i satisfies U_(i)∈({U₁, . . . U_(n)}−R)−{U₁}} and the value k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(j), . . . , T_(i−1) with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R (here, T_(j) is nil with respect to j which satisfies U_(j)∈R), transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}); a first session key generation step in which the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}) compute the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l) and compute k₁|s₁ by an exclusive OR of the values T′ and K₁ ^(l); and a second session key generation step in which the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) generate the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function.
 6. The key exchange method according to claim 5, in which time denotes current time, ID denotes a predicate variable representing the communication device, and TF denotes a predicate variable representing a time frame of the communication device, and a master secret key msk for functional encryption and a common key K₁ which is generated in a session established by the communication devices U₁, . . . , U_(n) are further stored in the storage of the key distribution device S, the key exchange method further comprising: a user key transmission step in which the key distribution device S generates a user secret key usk_(i) by using the master secret key msk with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R with attribution used as A_(i)=(U_(i), time) based on a key derivation algorithm for functional encryption, and encrypts the user secret key usk_(i) by using a public key pk_(i) of the communication devices U_(i) based on an encryption algorithm for public key encryption so as to generate a cipher text CT_(i); and a user key reception step in which the communication devices U_(i)∈{U₁, . . . , U_(n)}−R decrypt the cipher text CT_(i) by using the secret key sk_(i) based on a decryption algorithm for public key encryption so as to obtain the user secret key usk_(i), wherein in the third key generation step, in a case where a time frame has been changed from the session established by the communication devices U₁, . . . , U_(n), the key distribution device S generates the common key K₁ by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, and in a case where a time frame has not been changed from the session established by the communication devices U₁, . . . , U_(n), the key distribution device S reads the common key K₁ stored in the storage and encrypts the common key K₁ with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R with an access structure P_(i)=(ID=U_(i))∧(time∈TF) based on an encryption algorithm for functional encryption so as to further generate a cipher text CT′_(i), and in the second session key generation step, the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) decrypt the cipher text CT′_(i) by using the user secret key usk_(i) based on a decryption algorithm for functional encryption so as to obtain the common key K₁ and further compute a session key SK by an exclusive OR of a value generated by using (sid, K₁) based on the pseudo-random function and a value generated by using (sid, K₂) based on the pseudo-random function.
 7. The key exchange method according to claim 6, in which in the user key transmission step, the key distribution device S further generates a MAC key mk_(i) based on a key generation algorithm for a message authentication code and encrypts the user secret key usk_(i) and the MAC key mk_(i) by using the public key pk_(i) of the communication devices U_(i) based on the encryption algorithm for public key encryption so as to generate the cipher text CT_(i), and in the user key reception step, the communication devices U_(i)∈{U₁, . . . , U_(n)}−R decrypt the cipher text CT_(i) by using the secret key sk_(i) based on the decryption algorithm for public key encryption so as to obtain the user secret key usk_(i) and the MAC key mk_(i), the key exchange method further comprising: a first authentication tag generation step in which the representative communication device U₁ generates an authentication tag σ₁ by using the MAC key mk₁ and the values R₁, c₁, (R₃, R_(n−1)), T₁, T′, U₁, and sid based on the tag generation algorithm for a message authentication code, the communication devices U_(i) (i satisfies U_(i)∈N and U_(i+1) ∈R) generate an authentication tag σ_(i) by using the MAC key mk_(i) and the values R_(i), c_(i), R_(j), k_(i), s_(i), T_(i), U_(i), and sid based on the tag generation algorithm for a message authentication code, the communication devices U_(i) (i satisfies U_(i) ∈N and U_(i−1)ΣR) generate an authentication tag σ_(i) by using the MAC key mk_(i) and the values R_(i), c_(i), R_(j′), k_(i), s_(i), T_(i), U_(i), and sid based on the tag generation algorithm for a message authentication code, and the communication devices U_(i) (i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N) generate an authentication tag σ_(i) by using the MAC key mkd, and the values c_(i), k_(i), s_(i), T_(i), U_(i) and sid based on the tag generation algorithm for a message authentication code; a first authentication tag verification step in which the key distribution device S receives (T₁, T′, σ₁) from the representative communication device U₁, receives (k_(i), s_(i), T_(i), σ_(i)) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), verifies the authentication tag σ_(i) by using the MAC key mk_(i) (i satisfies U_(i)∈{U₁, . . . , U_(n)}−R) based on a verification algorithm for a message authentication code, and verifies whether or not c_(i)=g^(ki)h^(si) is satisfied with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}; a second authentication tag generation step in which the key distribution device S generates an authentication tag σ′₁ with respect to i=1 by using the MAC key mk₁ and the values R₁, c₁, (R₃, R_(n−1)), T₁, T′, U₁, sid, k′, and CT′₁ based on the tag generation algorithm for a message authentication code, generates an authentication tag σ′_(i) with respect to i which satisfies U_(i) ∈N and U_(i+1)∈R by using the MAC key mk_(i) and the values R_(i), c_(i), R_(j), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, and CT′_(i) based on the tag generation algorithm for a message authentication code, generates an authentication tag σ′_(i) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R by using the MAC key mk_(i) and the values R_(i), c_(i), R_(j′), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, and CT′_(i) based on the tag generation algorithm for a message authentication code, and generates an authentication tag σ′_(n−1) with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N by using the MAC key mk_(n−1) and the values c_(i), k_(i), s_(i), T_(i), U_(i), sid, c₁, k′, T′_(i), T′, and CT′_(i) based on the tag generation algorithm for a message authentication code; and a second authentication tag verification step in which the representative communication device U₁ receives (k′, CT′₁, σ′₁) from the key distribution device S and verifies the authentication tag σ′₁ by using the MAC key mk₁ based on the verification algorithm for a message authentication code, and the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}) receive (c_(i), k′, T′_(i), T′, CT′_(i), σ′_(i)) from the key distribution device S, verify the authentication tag σ′_(i) by using the MAC key mk_(i) based on the verification algorithm for a message authentication code, and verify whether or not c₁=g^(k1)h^(s1) is satisfied.
 8. A key exchange system comprising: a key distribution device S; and n pieces (here, n is an integer which is 2 or larger) of communication devices U_(i) (i=1, . . . , n), wherein R={U_(j1), . . . , U_(jm)} is a subset of {U₁, . . . , U_(n)} and N={U_(j1−i), U_(j1+1), . . . , U_(jm−1), U_(jm1)} is a subset of {U₁, . . . , U_(n)}, ∥ is a concatenation operator, U₁ (∈N) is one piece of representative communication device which is selected from N, the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i), c_(i)) from the communication devices U_(i) (∈N), receives a value c_(i) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−N), generates a value sid by using {c_(i)|i satisfies U_(i)∈{U₁, . . . , U_(n)}−R} based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(j)) with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (here, j is a minimum index which satisfies U_(j)∈N and j>i), (sid, R_(j′)) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i), and the value sid with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes a value k′ by an exclusive OR of {k_(i)|i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}} and the value k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(j), T_(i−1) (here, T_(j) is nil with respect to j which satisfies U_(j)∈R) with respect to i which satisfies U₁∈{U₁, . . . U_(n)}−R, transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), and the communication devices U_(i) (∈{U₁, . . . U_(n)}−R) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and stores information H_(i) ^(l) and H_(i) ^(r) generated in a session established by the communication devices U₁, . . . , U_(n), circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i which satisfies U_(i)∈N, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, execute a second key generation processing which, with respect to i=1, receives (sid, R₃, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(j′)=U_(n−1) are satisfied, receives (sid, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, H₁ ^(r)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j′)=U_(n−1) and U₂∈N are satisfied, and receives (sid, R₃) from the key distribution device S, generates a value K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(n)∈N are satisfied; with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (i is not 1), receives (sid, R_(j)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, R_(j) ^(ri)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (i is not 1), receives (sid, R_(j′)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(j) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (i is not 1), receives the value sid from the key distribution device S, generates a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S, and execute a session key generation processing which, with respect to i=1, receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function, and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function.
 9. A communication device which is included in a key exchange system comprising: a key distribution device S; and n+k pieces (here, n is an integer which is 2 or larger and k is an integer which is 1 or larger) of communication devices U_(i) (i=1, . . . , n+k), wherein ∥ is a concatenation operator, U₁ is one piece of representative communication device which is selected from the communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k), the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i), c_(i)) from the communication devices U_(i) (i=1, n, . . . , n+k), receives a value c_(i) from the communication devices U_(i) (i=2, . . . , n−1), generates a value sid by using the values c₁, . . . , c_(n+k) based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(i−1)) with respect to i=1, 2, the value sid with respect to i=3, . . . , n−2, (sid, R_(i+1)) with respect to i=n−1, n, and (sid, R_(i−1), R_(i+1)) with respect to i=n+1, . . . , n+k (here, R₀=R_(n+K), R_(n+k+1)=R₁), and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication device U_(i) (i=2, n−1, . . . , n+k), receives (k_(i), s_(i)) from the communication devices U_(i) (i=3, . . . , n−2), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on a twisted pseudo-random function, computes a value k′ by an exclusive OR of the values k₂, . . . , k_(n+k), k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . T_(i−1), with respect to i=2, . . . , n+k (here, T_(i) is nil with respect to i=3, . . . n−1), transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (i=2, . . . , n+k), and the communication devices U_(i) (i=1, . . . , n+k) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and further stores information r generated in a session established by the communication devices U₁, . . . , U_(n) with respect to i=1, . . . , n, circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i=1, n, . . . , n+k, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i=2, . . . , n−1, execute a second key generation processing which receives (sid R_(n+k)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S with respect to i=1, receives (sid, R₁) from the key distribution device S, generates a value K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function, generates a value K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function, computes the value T₂ by an exclusive OR of the values K₂ ^(l) and K₂ ^(r), and transmits (k₂, s₂, T₂) to the key distribution device S with respect to i=2, receives the value sid from the key distribution device S and transmits (k_(i), s_(i)) to the key distribution device S with respect to i=3, . . . , n−2, receives (sid, R_(n)) from the key distribution device S, generates a value K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function, generates a value K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function, computes the value T_(n−1) by an exclusive OR of the values K_(n−1) ^(l) and K_(n−1) ^(r), and transmits (k_(n−1), s_(n−1), T_(n−1)) to the key distribution device S with respect to i=n−1, receives (sid, R_(n+1)) from the key distribution device S, generates a value K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function, generates a value K_(n) ^(r) by using (sid, R_(n+1) ^(rn)) based on the pseudo-random function, computes the value T_(n) by an exclusive OR of the values K_(n) ^(l) and K_(n) ^(r), and transmits (k_(n), s_(n), T_(n)) to the key distribution device S with respect to i=n, and receives (sid, R_(i−1), R_(i+1)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S with respect to i=n+1, . . . , n+k, and execute a session key generation processing which receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=1, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=2, n, . . . , n+k, and receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and g^(r), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=3, . . . , n−1.
 10. A communication device which is included in a key exchange system comprising: a key distribution device S; and n pieces (here, n is an integer which is 2 or larger) of communication devices U_(i) (i=1, . . . , n), wherein R={U_(j1), . . . , U_(jm)} is a subset of {U₁, . . . , U_(n)} and N={U_(j1-1), U_(j1+1), . . . , U_(jm-1), U_(jm1)} is a subset of {U₁, . . . , U_(n)}, ∥ is a concatenation operator, U₁ (∈N) is one piece of representative communication device which is selected from N, the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i), c_(i)) from the communication devices U_(i) (∈N), receives a value c_(i) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R−N), generates a value sid by using {c_(i)|i satisfies U_(i)∈{U₁, . . . , U_(n)}−R} based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(j)) with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (here, j is a minimum index which satisfies U_(j)∈N and j>i), (sid, R_(j′)) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i), and the value sid with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes a value k′ by an exclusive OR of {k_(i)|i satisfies U_(i)∈({U₁, . . . U_(n)}−R)−{U₁}} and the value k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(j), . . . , T_(i−1) (here, T_(j) is nil with respect to j which satisfies U_(j)∈R) with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R, transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), and the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and stores information H_(i) ^(l) and H_(i) ^(r) generated in a session established by the communication devices U₁, . . . , U_(n), circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i which satisfies U_(i)∈N, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, execute a second key generation processing which, with respect to i=1, receives (sid, R₃, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(j′)=U_(n−1) are satisfied, receives (sid, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, H₁ ^(r)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j′)=U_(n−1) and U₂∈N are satisfied, and receives (sid, R₃) from the key distribution device S, generates a value K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(i)=U₃ and U_(n)∈N are satisfied; with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (i is not 1), receives (sid, R_(j)) from the key distribution device S, generates a value by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(l) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (i is not 1), receives (sid, R_(j′)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, R_(j′) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R−N (i is not 1), receives the value sid from the key distribution device S, generates a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S, and execute a session key generation processing which, with respect to i=1, receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function, and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function.
 11. A non-transitory computer-readable storage medium which stores a program for causing a computer to operate as a communication device which is included in key exchange system comprising: a key distribution device S; and n+k pieces (here, n is an integer which is 2 or larger and k is an integer which is 1 or larger) of communication devices U_(i) (i=1, . . . , n+k), wherein ∥ is a concatenation operator, U₁ is one piece of representative communication device which is selected from the communication devices U₁, U_(n), U_(n+1), . . . , U_(n+k), the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i), c_(i)) from the communication devices U_(i) (i=1, n, . . . , n+k), receives a value c_(i) from the communication devices U_(i) (i=2, . . . , n−1), generates a value sid by using the values c₁, . . . , c_(n+k) based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(i−1)) with respect to i=1, 2, the value sid with respect to i=3, . . . , n−2, (sid, R_(i+1)) with respect to i=n−1, n, and (sid, R_(i−1), R_(i+1)) with respect to i=n+1, . . . , n+k (here, R₀=R_(n+K), R_(n+k+1)=R₁), and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication device U_(i) (i=2, n−1, . . . , n+k), receives (k_(i), s_(i)) from the communication devices U_(i) (i=3, . . . , n−2), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on a twisted pseudo-random function, computes a value k′ by an exclusive OR of the values k₂, . . . , k_(n+k), k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(i−1) with respect to i=2, . . . , n+k (here, T_(i) is nil with respect to i=3, . . . , n−1), transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (i=2, . . . , n+k), and the communication devices U_(i) (i=1, . . . , n+k) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and further stores information r generated in a session established by the communication devices U₁, . . . , U_(n) with respect to i=1, . . . , n, circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i), and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i=1, n, . . . , n+k, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i=2, . . . , n−1, execute a second key generation processing which receives (sid, R_(n+k)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n+k) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, g^(r1r)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S with respect to i=1, receives (sid, R₁) from the key distribution device S, generates a value K₂ ^(l) by using (sid, R₁ ^(r)) based on the pseudo-random function, generates a value K₂ ^(r) by using (sid, g^(r)) based on the pseudo-random function, computes the value T₂ by an exclusive OR of the values K₂ ^(l) and K₂ ^(r), and transmits (k₂, s₂, T₂) to the key distribution device S with respect to i=2, receives the value sid from the key distribution device S and transmits (k_(i), s_(i)) to the key distribution device S with respect to i=3, . . . , n−2, receives (sid, R_(n)) from the key distribution device S, generates a value K_(n−1) ^(l) by using (sid, g^(r)) based on the pseudo-random function, generates a value K_(n−1) ^(r) by using (sid, R_(n) ^(r)) based on the pseudo-random function, computes the value T_(n−1) by an exclusive OR of the values K_(n−1) ^(l) and K_(n−1) ^(r), and transmits (k_(n−1), s_(n−1), T_(n−1)) to the key distribution device S with respect to i=n−1, receives (sid, R_(n+1)) from the key distribution device S, generates a value K_(n) ^(l) by using (sid, R_(n) ^(r)) based on the pseudo-random function, generates a value K_(n) ^(r) by using (sid, R_(n+1) ^(rn)) based on the pseudo-random function, computes the value T_(n) by an exclusive OR of the values K_(n) ^(l) and K_(n) ^(r), and transmits (k_(n), s_(n), T_(n)) to the key distribution device S with respect to i=n, and receives (sid, R_(i−1), R_(i+1)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, R_(i−1) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, R_(i+1) ^(ri)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S with respect to i=n+1, . . . , n+k, and execute a session key generation processing which receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=1, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=2, n, . . . , n+k, and receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and g^(r), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function with respect to i=3, . . . , n−1.
 12. A non-transitory computer-readable storage medium which stores a program for causing a computer to operate as a communication device which is included in key exchange system comprising: a key distribution device S; and n pieces (here, n is an integer which is 2 or larger) of communication devices U_(i) (i=1, . . . , n), wherein R={U_(j1), . . . , U_(jm)} is a subset of {U₁, . . . , U_(n)} and N={U_(j1-1), U_(j1+1), . . . , U_(jm-1), U_(jm1)} is a subset of {U₁, . . . , U_(n)}, ∥ is a concatenation operator, U₁ (∈N) is one piece of representative communication device which is selected from N, the key distribution device S includes a storage which stores a secret key sk_(S) for public key encryption and secret strings st_(S) and st′_(S) of the key distribution device S, circuitry configured to: execute a session ID generation processing which receives (R_(i); c_(i)) from the communication devices U_(i) (∈N), receives a value c_(i) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−N), generates a value sid by using {c_(i)|i satisfies U_(i)∈{U₁, . . . , U_(n)}−R} based on a target-collision resistant hash function, and transmits, to the communication devices U_(i), (sid, R_(j)) with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (here, j is a minimum index which satisfies U_(j)∈N and j>i), (sid, R_(j′)) with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (here, j′ is a maximum index which satisfies U_(j′)∈N and j′<i), and the value sid with respect to i which satisfies U_(i)∈({U₁, . . . U_(n)}−R)−N, and execute a third key generation processing which receives (T₁, T′) from the representative communication device U₁, receives (k_(i), s_(i), T_(i)) from the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), generates a value k_(s) by using the secret strings st_(S) and st′_(S) based on the twisted pseudo-random function, computes a value k′ by an exclusive OR of {k_(i)|i satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}} and the value k_(s), computes a value T′_(i) by an exclusive OR of the values T₁, . . . , T_(j), . . . , T_(i−1) (here, T_(j) is nil with respect to j which satisfies U_(j)∈R) with respect to i which satisfies U_(i)∈{U₁, . . . , U_(n)}−R, transmits the value k′ to the representative communication device U₁, and transmits (k′, T′_(i), T′) to the communication devices U_(i) (∈({U₁, . . . , U_(n)}−R)−{U₁}), and the communication devices U_(i) (∈{U₁, . . . , U_(n)}−R) include a storage which stores a secret key sk_(i) for public key encryption and secret strings st_(i) and st′_(i) of the communication devices U_(i) and stores information H_(i) ^(l) and H_(i) ^(r) generated in a session established by the communication devices U₁, . . . , U_(n), circuitry configured to: execute a first key generation processing which generates values r_(i), k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on a twisted pseudo-random function, computes the values R_(i)=g^(ri) and c_(i)=g^(ki)h^(si) (here, each of g and h denotes a generation source of a group G, which is a multiplication cyclic group of a prime number order p of κ bits), and transmits (R_(i), c_(i)) to the key distribution device S with respect to i which satisfies U_(i)∈N, and generates values k_(i) and s_(i) by using the secret strings st_(i) and st′_(i) based on the twisted pseudo-random function, computes the value c_(i)=g^(ki)h^(si), and transmits the value c_(i) to the key distribution device S with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N, execute a second key generation processing which, with respect to i=1, receives (sid, R₃, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by an exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by an exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(j′)=U_(n−1) are satisfied, receives (sid, R_(n−1)) from the key distribution device S, generates a value K₁ ^(l) by using (sid, R_(n−1) ^(r1)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, H₁ ^(r)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j′)=U_(n−1) and U₂∈N are satisfied, and receives (sid, R₃) from the key distribution device S, generates a value K₁ ^(l) by using (sid, H₁ ^(l)) based on the pseudo-random function, generates a value K₁ ^(r) by using (sid, R₃ ^(r1)) based on the pseudo-random function, computes the value T₁ by the exclusive OR of the values K₁ ^(l) and K₁ ^(r), computes the value T′ by the exclusive OR of the value K₁ ^(l) and k₁∥s₁, and transmits (T₁, T′) to the key distribution device S in a case where U_(j)=U₃ and U_(n)∈N are satisfied; with respect to i which satisfies U_(i)∈N and U_(i+1)∈R (i is not 1), receives (sid, R_(j)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, R_(j) ^(ri)) based on the pseudo-random function, computes the value T_(i) by an exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; with respect to i which satisfies U_(i)∈N and U_(i−1)∈R (i is not 1), receives (sid, R_(j′)) from the key distribution device S, generates a value K_(i) ^(l) by using (sid, R_(j′) ^(ri)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S; and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−N (i is not 1), receives the value sid from the key distribution device S, generates a value K_(i) ^(l) by using (sid, H_(i) ^(l)) based on the pseudo-random function, generates a value K_(i) ^(r) by using (sid, H_(i) ^(r)) based on the pseudo-random function, computes the value T_(i) by the exclusive OR of the values K_(i) ^(l) and K_(i) ^(r), and transmits (k_(i), s_(i), T_(i)) to the key distribution device S, and execute a session key generation processing which, with respect to i=1, receives the value k′ from the key distribution device S and generates a common key K₂ by using the value sid and an exclusive OR of the values k′ and k₁ based on the pseudo-random function, and with respect to i which satisfies U_(i)∈({U₁, . . . , U_(n)}−R)−{U₁}, receives (k′, T′_(i), T′) from the key distribution device S, computes the value K₁ ^(l) by an exclusive OR of the values T′_(i) and K_(i) ^(l), computes k₁∥s₁ by an exclusive OR of the values T′ and K₁ ^(l), and generates the common key K₂ by using the value sid and the exclusive OR of the values k′ and k₁ based on the pseudo-random function. 